Clarifying The Bent Coin Problem [Archive] - Two Plus Two Newer Archives

admiralfluff

06-03-2007, 05:39 AM

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I thought you were talking about a a bent coin, not a double headed one. My mistake. Bent coin to me means "not a fair flip but tails is still possible".

[/ QUOTE ]

A double-headed coin is just the most extreme bent coin. There is no discontinuity. Imagine a 99% bent coin. For a single choice strategy, the distribution looks a lot like the 2 spike 'double-headed', but each cluster is normal, and has tails.

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Well that's exactly the same as betting $1,000,000 on a single flip. It's not really a million flips.

[/ QUOTE ]

for a fully bent, or 'double headed coin' this is true. For bends <1, it is not. Again consider the 99% bend. You are very likely to be either very close to 1000000 wins, or very close to 0 wins, but it is no longer a guarantee.

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And you can get the same outcome as the fair coin by randomly choosing heads or tails on a biased coin.

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I might have to do some math to confirm this one for myself, but this sounds reasonable.

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So I don't understand what you're saying.

[/ QUOTE ]

Simply this:

For a single flip, there is no difference between an unknown biased coin, and a fair coin. For a series of flips, there is. The key is that we have control over the nature of our results distribution, per our flipping strategy.

Example game:
You have $2. Your opponent gives you 11:10 per flip of a coin. The catch: you must give your money to a 3rd party, and flip 100 times in a row. You will not see the results of your flip. The 3rd party will. After you have completed your 100 flips, the opponent will sequentially pay or rake from your roll based on your sequence of flips. If you go bust before going through all 100 flips, game over, the rest don't count. Your opponent offers either a fair coin, or a coin of unknown bias. Which coin should you choose, and what should your flipping strategy be?

wtfsvi

06-03-2007, 07:36 AM

[ QUOTE ]
That is not the same as saying there is a 50% chance the bent coin will come up heads. DUCY? If not, see my last post on your original thread for this topic.

[/ QUOTE ] Consider this scenario. You're playing texas holdem, and on the turn there are two clubs on the board and your two hole cards are clubs. The top card in the deck will be dealt for the river. This card is either a club or not a club. If we knew the card, there's either 100% chance you will hit your flush or 0%. Yet I know nothing about the card to come, and assuming it's a fair game I will say there is a 19% chance you will hit your flush.

This seems to me to be the same scenario as the bent coin. (And indeed better, because the 50% number is root for confusion, since when it's 50% it does not matter what side you pick and any bet will be +EV given 11 to 10.) Now it is obviously +EV to bet as if you have 19% chance to hit your flush. But do you disagree that you have a 19% chance of hitting the flush on the river?

edit: If you do, I think this is just a nitty discussion where you and Jason will not allow us to say the probability of event x is y% unless we know absolutely everything that might affect the event. In stead we will have to use the term "our best estimation of the probability is" every time we talk about probability in a practical context, and "the probability is" will be reserved for only the most abstract theoretical contexts. And that just seems really nitty.

David Sklansky

06-03-2007, 12:24 PM

[ QUOTE ]
[ QUOTE ]
Am I right to assume they disagree with the following statement

[/ QUOTE ]
No.

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Well as far as I'm concerned that is all I am saying. So I don't fully understand what you are arguing about. Neither does anyone else here. I would also like to ask again whether you think Persi Diaconis agrees with you.

ALawPoker

06-03-2007, 12:47 PM

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Doing that I have a 50-50 chance of winning the bet. I'll take the 12-10 odds on the bent coin.

That is not the same as saying there is a 50% chance the bent coin will come up heads.

[/ QUOTE ]

Correct me if I'm wrong, but it seems ALL you and Jason are arguing is that there doesn't *truly* exist a 50% chance that the coin will land on heads, even though assuming there does is still the best way to approach decisions. I don't think Sklansky or anyone disagrees with that.

To go back to the AA vs. KK example (with a fair shuffle) you again are either a 100% or 0% shot to win when the money gets in. You just have no way of knowing how variance (or the Poker Gods, if you're a religious person) bent the board cards. You have no way of *truly* knowing any better, so you assume the 88% as the fact of the matter. It really is the same thing.

Gah, this discussion is bizarre.

ALawPoker

06-03-2007, 01:00 PM

Actually I just want to ask this, to PTB and Jason:

So, there isn't *really* a 50% shot it will land on heads, but you both seem to agree you would make decisions by assuming there is. So, what use is it? What sort of reality is worth considering when the best thing to do is ignore it?

If aliens exist on a far away planet, and I (or anyone I care about) will never see them, to me they don't exist. There are a lot of things that exist beyond the scope of our capacity to understand them. But then, what use is that? It seems life is more about properly interpreting and reacting to our environment; and not about worrying what does and does not exist in the eyes of omniscient understanding.

slickpoppa

06-03-2007, 01:15 PM

Off topic, but here is an interesting Ken Uston coin flipping story:

[ QUOTE ]
Snyder: The story goes like this: In an elevator, sometimes a parking lot, you got into a coin flipping, or a coin tossing contest with someone, and lost a lot of money. You then tried to get your blackjack team to pay for the money you lost out of the team bankroll, because had you won, you allegedly assured your teammates, you would have put your winnings into the team bankroll.

Uston: That's absolutely true.

Snyder: Well, maybe you can fill in some of the details. You've said that you never gamble, that you are an investor, and that you only risk money on positive expectation ventures. How do you justify a contest like this as a positive expectation gamble?

Uston:I was playing at the Holiday Inn. I remember driving down the Strip thinking, "Where the hell am I going to play?" I was feeling very paranoid at the time about the fact that I wasn't contributing to the team the way I should be, because of the fact that I couldn't play very many places. Somehow, I sauntered into the Holiday, and I got a game at the single deck table there. You know the one - the one that's colored red instead of green. And, I'm sitting at the table and playing - I don't remember if I was winning or losing but there was this crazy guy at third base, a big fat guy. He's talking and playing, and obviously recognizes me, but the people in the pit don't. At some point he comes over and sits next to me. He flipped a coin and he put it underneath a dollar bill. He said to me if I can guess what it is he'll give me . . . I think $500 - I can't remember the numbers - but if I guess wrong, I give him $100. He's a crazy guy. He just lost about $2000 over the third base. He's a terrible player, just throwing his goddamn money around. And I'm saying to myself, "Here I am playing through all this [censored] it was a full table - waiting through all this [censored], waiting for a 2% edge. And this guy gives me an edge of . . . whatever the figure was. And I looked at him at first and said, "What?" And he meant it. So, I said "Okay, Heads." And I lost and I gave him a hundred bucks. He's a very good con. He does this for a living. His problem is he's an inveterate gambler. He's told me, and I fully believe him, that he's made two or three hundred thousand dollars doing this at various places around - race tracks is one place that he particularly does this.

Anyway, I left the table at that point, really fascinated with this thing. God, what the hell is going on? So we went to the bar and had a drink, then we went back to the Jockey Club. I invited him back. Initially, we were going to go to the Aladdin and have a drink, but at the last minute, I said (snaps fingers), "Let's go to the Jockey Club." That's a significant factor. We walked into the Jockey Club bar, and we're sitting there again, and he's good with the con, saying, "Ken, I don't want to do it again. You're too nice a guy." Naturally, he's sucking me in beautifully. So what he does the next time, he has it where these people sitting around the Jockey Club bar are all my friends. We came at the last minute. There could be nobody there he knew. There couldn't have been any way he knew we were going there. And he says to someone in the bar, "Why don't you flip the coin, and you call it. And if you're right, I'll give Kenny $800. But if you're wrong, Kenny's got to give me $100." And I'm thinking, "This guy's crazy." And I want to get my hundred bucks back. There's some con in me, too, sure. And I go along with it. And I lose. And then he offers me greater odds, to the extent that I finally end up losing just under $10,000 to him. I think it was $9,400.

So, at the very end, to get the nine grand to give him - I mean, it's not a lot of money. We're playing off a $100,000 bank. But, I lose $9,400, and I have to go to my safe deposit box at the front of the Jockey Club to get the money out of my box. Now, get this bit. This is incredible. He says, "I'll tell you what, Kenny. I don't want to take your money. You're too good a guy, really." All the rap, he goes on and on and on and on. And he turns to the clerk at the desk. May God be my witness, all this is totally true. He says to the clerk at the desk, "You flip the coin, and if the bellman calls it right, I'll call off the S9,400. And if he doesn't call it right, Kenny, you pay me $9,400. He's giving me a $9,400 to 0 bet. And I lost. And I gave him the money. I told the team about this, that I lost a total of $9,400. The first thing I did, I ran up to one of our rooms, and I said, "You would not believe I got a 90% edge over this guy!" And I explained what was going on, and they were all very suspicious. I'm saying, "No, you've got to see this!" But what happened was, to make a long story short, we had a meeting to determine whether or not it should come out of the team money, and the net result was that it didn't.

Snyder: That was just the way I heard it. They refused to cover your loss.

Uston: I took a polygraph on it. They were worried about the whole issue, why I'm out there flipping coins, and that the extent of the loss was exactly $9,400. We had a big team meeting, and a discussion, and they said, "No, it can't come out of the team money. It's got to come out of your money."

Snyder: Do you know how the con worked?

Uston: Yes, I met the guy. He came back a little later. He stayed away from me for a while because, he thought, with me being a big gambler and all that, I was going to get the mob after him. But he finally came back about 3 or 4 months later. To that day, 3 or 4 months later, I was convinced I'd had an edge over this man. There was no way - I wasn't flipping the coin, he wasn't flipping the coin, I wasn't calling it, somebody else was . . . Two totally different people! Well, he came back and he explained the way it worked. First of all, he said that he has very quick eyes, and he can flip a coin to land any way he wants - which is totally irrelevant because he wasn't flipping the coin. But because of his ability to see coins, he knew what the coins were when they were flipped by another person. And he said he was uncannily lucky that night. Eight out of ten times he won the bet legitimately. There were a couple times when he didn't, and what he did was somehow talk the person out of it. The way he did it, if this guy said "Heads," he'd say, "You sure you want to make it heads? You don't want to make it tails?" In other words, after the other guy guessed it, he would engage in a little rap for a while, and either he'd talk the person out of it, or increase the odds and have another flip. Somehow, by doing that, and being able to know whether the person was right or wrong, plus having the correct thing going for him 8 out of 10 times anyway, he totally pulled the wool over my eyes. That's how he did it. It's so funny, because later in Atlantic City he lured another team member of mine. I won't tell you his name, but it was December of '79. Anyway, the story's true.

[/ QUOTE ]

chezlaw

06-03-2007, 01:22 PM

[ QUOTE ]
Actually I just want to ask this, to PTB and Jason:

So, there isn't *really* a 50% shot it will land on heads, but you both seem to agree you would make decisions by assuming there is. So, what use is it? What sort of reality is worth considering when the best thing to do is ignore it?

[/ QUOTE ]
Sorry, chez here. It's not right to assume there's a 50% chance that coin will land on heads - everyone agrees there's some decisions where it would be very -ev to assume that. However there are some decisions where its 50:50 even if the coin definitely doesn't have a 50:50 chance to land on heads e.g. when you know the coin is biased 75:25 but there's no information which way its biased.

As DS is only talking about the decisions where it makes no difference the whole discussion seems pointless and I've still no idea what difference it makes to the jurors thread.

chez

PairTheBoard

06-03-2007, 01:23 PM

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[ QUOTE ]
That is not the same as saying there is a 50% chance the bent coin will come up heads. DUCY? If not, see my last post on your original thread for this topic.

[/ QUOTE ]

Consider this scenario. You're playing texas holdem, and on the turn there are two clubs on the board and your two hole cards are clubs. The top card in the deck will be dealt for the river. This card is either a club or not a club. If we knew the card, there's either 100% chance you will hit your flush or 0%. Yet I know nothing about the card to come, and assuming it's a fair game I will say there is a 19% chance you will hit your flush.

This seems to me to be the same scenario as the bent coin.

[/ QUOTE ]

The difference has been explained on the original Thread, both by Jason and myself.This is why the Forum considers it bad form to do as David commonly does and split a topic up into more than one thread. It makes it hard to follow the discussion and leads to duplication of effort.

With the cards, the assumptions are clear and the experiment can be repeated in practice. The deck has been shuffled. We can repeat the experiment in practice.

With the coin, the assumptions are about an imaginary world full of bent coins where we assume the biases created by the bends have some kind of distribution for the biases which allow your conclusion. That's a lot of hidden asssumptions. They cannot be tested. It's an imaginary world. That imaginary world is not available to us. Only the coin in front of us is available.

See also my post on the other thread about the Two Envelope problem. It forcefully illustrates the kind of trouble you can get into by making this kind of vague mathematically undefined statement about the probability the bent coin comes up heads being 50%.

You might think this is being nitty. It's not. This is why we are so careful doing mathematics. The Vauge seat of the pants type stuff that Sklansky insists on doing here can lead to a lot of trouble.

PairTheBoard

jason1990

06-03-2007, 01:35 PM

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So I don't fully understand what you are arguing about.

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Obviously.

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Neither does anyone else here.

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Wrong.

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I would also like to ask again whether you think Persi Diaconis agrees with you.

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Not enough information.

I would also like to ask something again:

--------------------
QUOTE:
--------------------
Okay, look, I just bent a real penny. It is sitting here on my desk. Here are two questions:

(a) What is the probability the penny comes up heads?
(b) What is the probability the penny comes up tails?

Pick a question, either question, and answer it. You can answer with "not enough information" or you can answer with a number between 0 and 1.

All of your posts seem to indicate that your answer to both questions is 0.5. Yet I do not think you have come right out and said that. Why? Am I misunderstanding you?
--------------------

PairTheBoard

06-03-2007, 01:55 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Am I right to assume they disagree with the following statement

[/ QUOTE ]
No.

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Well as far as I'm concerned that is all I am saying. So I don't fully understand what you are arguing about. Neither does anyone else here. I would also like to ask again whether you think Persi Diaconis agrees with you.

[/ QUOTE ]

Where do you think he agrees with you? Since he's not here to speak for himself do you intend to represent his views? Do you assert that your views represent his? Where are your sources for such a contention? Any quotes?

If you are as MSL illiterate as you are beginning to appear, I wonder if you could even understand what Persi Diaconis might say if he were here to speak.

PairTheBoard

wtfsvi

06-03-2007, 02:10 PM

[ QUOTE ]
With the cards, the assumptions are clear and the experiment can be repeated in practice. The deck has been shuffled. We can repeat the experiment in practice.

With the coin, the assumptions are about an imaginary world full of bent coins

[/ QUOTE ] Yes. Is there something wrong with this imaginary world full of bent coins?

[ QUOTE ]
where we assume the biases created by the bends have some kind of distribution for the biases which allow your conclusion.

[/ QUOTE ] ? No. What are you talking about? That there might be some factor that makes coin-benders more likely to bend coins in favor of heads? Sure, if we knew that, that would make a difference. Such a bias probably exists to one side or the other, but if you can't tell me what side you think it is it will have no bearing on the probability model you should base your actions on.

As for the envelope problem, that is quite interesting. I can't quite wrap my head around it, so that might explain why I can't wrap my head around what you are trying to say. I will read up on it.

PairTheBoard

06-03-2007, 02:33 PM

[ QUOTE ]
[ QUOTE ]
where we assume the biases created by the bends have some kind of distribution for the biases which allow your conclusion.

[/ QUOTE ]

? No. What are you talking about? That there might be some factor that makes coin-benders more likely to bend coins in favor of heads? Sure, if we knew that, that would make a difference. Such a bias probably exists to one side or the other, but if you can't tell me what side you think it is it will have no bearing on the probability model you should base your actions on.

[/ QUOTE ]

What actions? I already know what my actions are going to be based on a sound mathematical model. I'm going to flip a fair coin to choose my Call and call the bent coin in the air. I need no imaginary world to form this model for this particular coin and know I will have a 50-50 chance of making the winning Call.

That's different than making the Vague mathematically ill-defined statement, P(Heads) = 50%. For that statement to have mathematical validity you would need to look at the imaginary world I described. Furhermore, as you point out, it's very possible there will be a bias to how coins might get bent in that imaginary world. Especially if the world happens to consist of one bent coin being bent by the guy offering to let you bet on heads. But even supposing he is just randomly part of this larger imaginary world were there's a bias to how the coins get bent, you then have to look at a universe of imaginary pre-worlds where the probabilities for bending-biases come. All this to make the stament P(Heads) = 50% meaningful. All those assumptions about imaginary worlds which can't be tested make the stament P(heads)=50% Subjective, not mathematical.

Sometimes it might be useful to make such a subjective assesment. Other times it can get you into trouble. Especially if you are not aware of what you are doing. In the case of the suffled cards it's easy to be aware of your assumptions about the suffle, and just to be sure you can practically test them. Not the case with the bent coin and not the case with a lot of other statements like this that Sklansky asserts by the seat of his pants.

When you allow yourself to habitually think in such a mathematically imprecise way you are just asking for the kind of trouble you get into in the Two Envelope Problem.

PairTheBoard

wtfsvi

06-03-2007, 02:39 PM

So can you give me a probability statement about anything in a real world context that is not subjective?

edit: Each shuffle situation will be different. The deck had a different makeup to start the shuffle. Different dealer. Different a million different things. You can not run an identical scenario several times to test the probability. You never can.

Double edit: And if you could test the poker example several times in the exact same environment, you would come to the shocking conclusion that hitting a flush on the river is either 100% or 0% likely.

David Sklansky

06-03-2007, 02:39 PM

My point is that there is no such thing as THE probability of an event. There is only a probability correlated with the information you have.

In the specific case where the only information is the number of choices than the probability associated with that information has historically been one over that number.

And I assume that you would agree that if Diaconis disagreed with you, that would at the very least make it reasonable to disagree with you. (I have no idea if he does.)

David Sklansky

06-03-2007, 02:43 PM

"This is why the Forum considers it bad form to do as David commonly does and split a topic up into more than one thread. It makes it hard to follow the discussion and leads to duplication of effort."

I do that when it looks like most people have left the thread and I want to bring them back.

admiralfluff

06-03-2007, 03:12 PM

[ QUOTE ]
But do you disagree that you have a 19% chance of hitting the flush on the river?

[/ QUOTE ]

Of course you have a 19% chance of hitting your flush. This is not my point at all. My point is only concerning a series of trials. In the context of your example:

Assume you have a stacked deck, of unknown bias, a fd, and need to hit a club. The deck is stacked so that the club will hit, or not, you don't know which. For one trial, this is identical to playing with a unknown deck. But, if you are allowed to run it twice, or ten times, it matters. If the deck is stacked so you will always hit a club, or always not, you can control the probability distribution of your outcomes to alter variance, but not EV. See my eariler posts if you don't understand how this works.

PairTheBoard

06-03-2007, 03:13 PM

[ QUOTE ]
And I assume that you would agree that if Diaconis disagreed with you, that would at the very least make it reasonable to disagree with you. (I have no idea if he does.)

[/ QUOTE ]

Then why bring it up? I recognize he is a Super-Expert on Probability. Since neither of us have read him we can't know for sure what he says. Are you implying there is therefore a 50% chance he will disagree with me? That's exactly the kind of nonsense your Sklanksy-Probability-Logic can produce.

The fact is that I've studed the mathematics of probability extensively. You haven't. I didn't study under Diaconis but I did study under people who are also Super-Experts in the Field. You haven't. jason1990 is a real expert and I'm saying nothing different than what he is saying. You on the other hand are an amateur probablist with a high IQ who knows enough probability to write good poker books. But on this question you are flying by the seat of your pants and beginning to look more and more like the person you described here,

[ QUOTE ]
DS -
they think that because they have above average IQs they shouldn't be considered morons when they offer their opinions about stuff that isn't obviously highly mathematical. When they enounter a subject that is 20% mathematical they either deny that it percentage, or claim that they can overcome the 20%. Thus they are in fact morons.

[/ QUOTE ]

This situation is even worse. This is a question that is almost 100% mathematical yet you think your high IQ can make up for your lack of mathematical knowledge.

PairTheBoard

wtfsvi

06-03-2007, 03:22 PM

I'm not disagreeing with you. Only pairtheboard and jason I think. Of course the 2nd flip of the bent coin is not 50% likely to be heads after we know what the outcome of the first flip was.

PairTheBoard

06-03-2007, 03:31 PM

[ QUOTE ]
So can you give me a probability statement about anything in a real world context that is not subjective?

edit: Each shuffle situation will be different. The deck had a different makeup to start the shuffle. Different dealer. Different a million different things. You can not run an identical scenario several times to test the probability. You never can.

Double edit: And if you could test the poker example several times in the exact same environment, you would come to the shocking conclusion that hitting a flush on the river is either 100% or 0% likely.

[/ QUOTE ]

The objective probablity statement that can be made is one that says, This Model computes this probability. Its accuracy for describing real world events depends on the accuracy of its assumptions about the preconditions for those events. In the case of the suffle the preconditions can be statistically tested to see how accurate they are. In the case of the bent coin, they can't.

We make probability statements in that way so that we can be clear on what we mean by them. Whether or not the current deck of cards actually meets the preconditions we define and test is a subjective question we cannot answer objectively. So the probability statement you insist on making is indeed subjective. You make it at your own peril. You can try to understand this or continue with sloppy thinking and never know for sure where you stand with your later conclusions. You never know when you are going to end up making a foolish conclusion and decide to pay the 10% to switch envelopes.

PairTheBoard

admiralfluff

06-03-2007, 03:35 PM

[ QUOTE ]
I'm not disagreeing with you. Only pairtheboard and jason I think. Of course the 2nd flip of the bent coin is not 50% likely to be heads after we know what the outcome of the first flip was.

[/ QUOTE ]

Yeah, one of the conditions is that we don't know the results of the series of flips till the end.

jason1990

06-03-2007, 03:37 PM

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My point is that there is no such thing as THE probability of an event.

[/ QUOTE ]
Okay, great. Thank you for clarifying. You are not alone in this belief. Many philosophers agree with you. The idea that all probabilities are subjective and relative to an observer is possibly as old as probability itself. But I would like to firmly nail this down. How exactly does that translate into an answer to my question:

[ QUOTE ]
Okay, look, I just bent a real penny. It is sitting here on my desk. Here are two questions:

(a) What is the probability the penny comes up heads?
(b) What is the probability the penny comes up tails?

Pick a question, either question, and answer it.

[/ QUOTE ]
Am I correct in assuming your response is: "There is no answer to question (a). THE probability of heads for your bent coin does not exist. Same for (b)."

[ QUOTE ]
And I assume that you would agree that if Diaconis disagreed with you, that would at the very least make it reasonable to disagree with you. (I have no idea if he does.)

[/ QUOTE ]
You do not need Persi in order to reasonably disagree with me. But you do need to understand me.

David Sklansky

06-03-2007, 04:28 PM

"Okay, great. Thank you for clarifying. You are not alone in this belief. Many philosophers agree with you."

In that case I am in trouble.

"You do not need Persi in order to reasonably disagree with me. But you do need to understand me"

Maybe so. But if Persi Diaconis disagrees with what you are saying, that means there is a reasonable chance you are wrong or that the issue is actually just a matter of opinion or semantics or whatever. In either case I would abandon the argument, resigned to the fact that you were used as a pawn to bolster a case that collaborating morons can usually be trusted to make good decisions.

chezlaw

06-03-2007, 04:30 PM

[ QUOTE ]
"Okay, great. Thank you for clarifying. You are not alone in this belief. Many philosophers agree with you."

In that case I am in trouble.

[/ QUOTE ]
not only a hi falootin philsopher but an idealistic one.

I'd have lost money on you being a realist.

chez

ALawPoker

06-03-2007, 04:59 PM

[ QUOTE ]
"This is why the Forum considers it bad form to do as David commonly does and split a topic up into more than one thread. It makes it hard to follow the discussion and leads to duplication of effort."

I do that when it looks like most people have left the thread and I want to bring them back.

[/ QUOTE ]

I think maybe locking the first thread might be a good idea when you do this.

David Sklansky

06-03-2007, 05:13 PM

"Then why bring it up? I recognize he is a Super-Expert on Probability. Since neither of us have read him we can't know for sure what he says. Are you implying there is therefore a 50% chance he will disagree with me? That's exactly the kind of nonsense your Sklanksy-Probability-Logic can produce.

The fact is that I've studed the mathematics of probability extensively. You haven't. I didn't study under Diaconis but I did study under people who are also Super-Experts in the Field. You haven't. jason1990 is a real expert and I'm saying nothing different than what he is saying. You on the other hand are an amateur probablist with a high IQ who knows enough probability to write good poker books. But on this question you are flying by the seat of your pants"

I am not going to try to figure the chances that Persi disagrees with you. But it isn't the 50-50 deal because we have more information. That's all irrelevant. I'm wondering about his stance because if he agrees with you I'll pay more careful attention to the details of what you are saying. If he disagrees it means you ideas have a good chance to be either flawed or just a matter of opinion.

T50_Omaha8

06-03-2007, 05:31 PM

So what's the probability of an asteroid hitting the Earth in the next 10,000 years?

No answer, LDO

Clearly this isn't a probability matter, since there either is an asteroid on a path that will inevitably lead it to collide with the Earth or there isn't. I can't believe the idoit mathematician who wrote my probability textbook attempts to model this event with an exponential distribution.

My understanding of probability has become way more solid since stumbling upon these threads...thanks

jason1990

06-03-2007, 05:51 PM

Thanks for the reply. Will you also be replying to this:

[ QUOTE ]
Am I correct in assuming your response is: "There is no answer to question (a). THE probability of heads for your bent coin does not exist. Same for (b)."

[/ QUOTE ]

PairTheBoard

06-03-2007, 06:36 PM

[ QUOTE ]
[ QUOTE ]
"Then why bring it up? I recognize he is a Super-Expert on Probability. Since neither of us have read him we can't know for sure what he says. Are you implying there is therefore a 50% chance he will disagree with me? That's exactly the kind of nonsense your Sklanksy-Probability-Logic can produce.

The fact is that I've studed the mathematics of probability extensively. You haven't. I didn't study under Diaconis but I did study under people who are also Super-Experts in the Field. You haven't. jason1990 is a real expert and I'm saying nothing different than what he is saying. You on the other hand are an amateur probablist with a high IQ who knows enough probability to write good poker books. But on this question you are flying by the seat of your pants"

[/ QUOTE ]

I am not going to try to figure the chances that Persi disagrees with you. But it isn't the 50-50 deal because we have more information. That's all irrelevant. I'm wondering about his stance because if he agrees with you I'll pay more careful attention to the details of what you are saying. If he disagrees it means you ideas have a good chance to be either flawed or just a matter of opinion.

[/ QUOTE ]

I can't imagine him saying anything of substance different than what jason or I am saying. You could try reading his papers and books. But then if you can't understand the mathematical principles jason and I are explaining to you I doubt you would understand them when Diaconis explains them, assuming he has spoken on this subject somewhere in his work.

The thing is David, you have never learned how to construct rigorous mathematical proofs. You have never spent the time to develop that skill. You have never worked in a wide range of mathematical subjects and learned to prove theorems by methods that would pass peer review. You have never worked on numerous problems requiring those kinds of proof. In fact, you have at times expressed an attitude of disdain for such conventional methods, relying instead on your natural untrained intelligence.

So it's not suprising that you are having trouble following what we are saying. You can drop the names of 100 highly respected mathematicians who you have not read and wonder what they might say. They are not here to speak. We are. If you had studied more mathematics you could follow what we are saying. Since you haven't you might at least listen and use that high IQ of yours to think about it.

PairTheBoard

djames

06-03-2007, 07:47 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Am I right to assume they disagree with the following statement

[/ QUOTE ]
No.

[/ QUOTE ]

Well as far as I'm concerned that is all I am saying. So I don't fully understand what you are arguing about. Neither does anyone else here. I would also like to ask again whether you think Persi Diaconis agrees with you.

[/ QUOTE ]

You are wrong about the statement in bold. I agree completely, as anyone training in mathematics and is familiar with probability would.

Why do you consistently question obvious intelligence by interrogating for educational background while namedropping when you yourself must hide behind an "I was too lazy" copout to explain why you don't have the credentials you require from posters that disagree with you?

I've lost a tremendous amount of respect for you over the past few weeks.

PairTheBoard

06-03-2007, 08:25 PM

I have read posts by djames here and in the Probablity Forum where he has done excellent high level mathematical work. He is well qualified to speak. Just an fyi for those who may not be aware of this fact.

PairTheBoard

ALawPoker

06-04-2007, 12:28 AM

What is it with the obsession with qualifications in this thread. Hurrrr I've studied this for 12 years I'm qualified this guy's qualified that guy isn't hurrrrr.

"I'm right because I know what I'm talking about" is ridiculously lame and a big sign of weakness, imo.

Whenever I read these threads I usually err on the side of rooting against Sklansky, if there's any reasonable debate. But this is just unfair. If you have a point to prove, use your expertise to prove it. This is a discussion forum, not an "I have more qualifications so listen to the truth because I say it's the truth" public service announcement.

PairTheBoard

06-04-2007, 01:05 AM

[ QUOTE ]
What is it with the obsession with qualifications in this thread. Hurrrr I've studied this for 12 years I'm qualified this guy's qualified that guy isn't hurrrrr.

"I'm right because I know what I'm talking about" is ridiculously lame and a big sign of weakness, imo.

Whenever I read these threads I usually err on the side of rooting against Sklansky, if there's any reasonable debate. But this is just unfair. If you have a point to prove, use your expertise to prove it. This is a discussion forum, not an "I have more qualifications so listen to the truth because I say it's the truth" public service announcement.

[/ QUOTE ]

Are you kidding? Sklansky's the one who started demanding qualifications. Sklansky's the one who demanded to know what schools people went to. Were they Harvard or Indiana? Sklansky's the one who brought Persi Diaconis into it even though he has no idea what Persi would say about the issue. He can't understand the math so he attacks the qualifications of the people trying to explain it to him. Maybe you're right. Maybe I should have taken the high road like others did and simply refuse to play his "qualifications" game. I'm really starting to wonder though if it's even worth participating in discussions with him at all.

PairTheBoard

Phil153

06-04-2007, 01:06 AM

[ QUOTE ]
You are wrong about the statement in bold. I agree completely, as anyone training in mathematics and is familiar with probability would.

[/ QUOTE ]
Do you care to spell it out in plain English? As far as I'm concerned, Jason is confusing the issues.

Leaky Eye

06-04-2007, 01:18 AM

I am glad you guys are so good at math. Too bad you haven't done much with reading comprehension. Sklansky chooses heads or tails randomly in his problem.

David Sklansky

06-04-2007, 01:45 AM

"I can't imagine him saying anything of substance different than what jason or I am saying. You could try reading his papers and books. But then if you can't understand the mathematical principles jason and I are explaining to you I doubt you would understand them when Diaconis explains them, assuming he has spoken on this subject somewhere in his work."

I picked Persi Diaconis to ask about because he is one of the best statisticians in the world, and because he is described as a "Baysian". And I am under the impression that Baysians are perhaps more likley to disagree with you. If he does agree with you I'll look at the details of what you are saying. If he doesn't, I see no reason to.

David Sklansky

06-04-2007, 01:57 AM

"Why do you consistently question obvious intelligence by interrogating for educational background while namedropping when you yourself must hide behind an "I was too lazy" copout to explain why you don't have the credentials you require from posters that disagree with you?"

If someone is disagreeing with me about something that I am almost sure about regarding mathematically related ideas, they are almost always wrong. If they back their disagreement up with techical arguments and they are obviously well educated and intelligent, I'll move them up to a small favorite over me. Unless they are extremely intelligent or well educated. Then I concede and perhaps study what they are saying. And since I actually know Persi Diaconis, I might be able to persuade him to look at these threads and post. If I get lucky he will agree with me and I can avoid some work.

ALawPoker

06-04-2007, 01:59 AM

[ QUOTE ]
Are you kidding? Sklansky's the one who started demanding qualifications.

[/ QUOTE ]

Hmm. I admit, I haven't read this whole thread. Maybe he did start it. But the stuff I read from him struck me more as that he was just curious. He disagreed with certain arguments in their own right, but in the instance of high qualifications, he would think twice.

Some of the things you were saying almost seemed like your qualifications were part of your argument, which imo is different than Sklansky's motivation for bringing it up.

I think maybe I've just gotten used to David's mindset. I can certainly see why it seems hypocritical of him to challenge people's credentials. If someone disagreed with something he said and demanded credentials before his argument would be considered further, I don't think he'd like it. Of course, in David's mind his argument would always be logically justifiable in its own right, so there would be no rational need for anyone to ever make such a demand *of him*. Hence, he doesn't recognize what he does as hypocrisy.

Whatever, it's not a big deal. This was an interesting thread though, and I just hate to see it cheapened. I guess that's the only reason I said anything. Nothing personal.

wtfsvi

06-04-2007, 02:18 AM

I think the essence of this is a question if you believe in absolute probability, like pairtheboard and jason, or don't, like sklansky. I can't imagine absolute probability without omniscience. (And with omniscience there is probably not much use for it /images/graemlins/tongue.gif) That makes it pretty useless to me to not be able to say "the probability is x%", when from my point of view the probability is x%. Of course, I can know, like in this case, that another point of view exists that is more accurate. And I can either act on the information I have from my perspective, or, if I think it's important, try to get the information necessary for the more accurate one.

TomCowley

06-04-2007, 02:51 AM

I'm not sure that the two envelopes problem is a very good rebuttal of "sklansky-esque" probabilistic thinking.

Let's modify the game a lot, defining a finite set of envelope pairs (2^X-1,2^X) with X as an integer on [1,Q]. This game is well defined, and switching is obviously 0EV over the sum of all possibilities. Now we can probabilistically model this game without dealing with any infinity-related messes.

If we choose an envelope pair at random, randomly open one of the envelopes, and then decide to switch, this is +EV in ALL cases except the case where 2^Q is opened. The case where 2^Q is opened is not in the spirit of the problem, because the original problem assumes that 2^Q+1 is possible. So, taking this finite snapshot, and calculating the EV of the switching, under the condition that with the chosen envelope, 2x and 0.5x envelopes are possible, switching is always +EV by the same fraction (as sklansky-esque thinking would predict)

Let's modify the game closer to the original formulation, but still with 0EV. Let X be an integer from (-inf,Q] with envelopes as before. You choose an envelope pair at random, open one at random, and find 2^X. Then you can either switch or not.

This game cannot be modeled by a uniform distribution P=constant of probabilities of envelope pairs (because there are an infinite number). If we take the standard assumption that all envelope pairs are equally likely, then even though an initial probability distribution cannot be defined, the expected value of the game over all switches is well defined (unlike the case where Q=inf) because it can be expressed by a convergent enumerated series and the expression of EV converges nicely to 0 (geometric series, I figure anybody who's interested in this can verify it), as it logically has to (the 2^Q forced downswitch wipes out the gain from all other switches). Using sklansky-esque thinking, it is always +EV to switch when 2x and .5x envelopes are possible, AND THE MATH BEARS THIS OUT WITH NO PARADOX.

Extended to the original problem, where Q->Inf, for any finite Q bigger than X (which is a base assumption of the problem, that double the amount is possible), it is mathematically well-defined that switching is a constant +EV fraction, even though a uniform distribution can't be used for standard probabilistic methods.

The "paradox" is the misapplication of the symmetry argument. The symmetry argument is that for a random envelope from a particular pair, switching must be 0EV. Derf. But you never have a random envelope from a particular pair. You either have the high envelope from X-1,X or the low envelope of X,X+1. The symmetry only applies to the two switches IN A PARTICULAR ENVELOPE PAIR, but the envelope pair is not defined until the second envelope's contents are known, at which point the symmetry argument doesn't start with a random envelope- it always starts with the low or the high (depending on the pair), so it is not logically sound.

This seems far too easy to me to be an unsolved paradox in mathematics (I guess if everybody was trying to prove switching=0EV, and it really isn't, there's a reason nobody's solved it), but I sure don't see what I'm missing.

TomCowley

06-04-2007, 03:43 AM

Or another way of looking at the paradox: Given a random envelope, the EV of switching is 0. Given a random envelope where 2x and 0.5x are possible, which you don't know until opening, the EV of switching is positive (because you can no longer hit the one giant negative). You aren't sampling the same set of envelopes. This can be translated to reals and the conditional switch EV can be defined as a limit as Q->inf without too much trouble to be completely equivalent to the original problem.

David Sklansky

06-04-2007, 03:58 AM

I'm not sure if the paradox relates to anything I've said or not. But even if it does, it is a contrived example involving numbers approaching infinity and PTB is implying that the indifference principle, or whatever it is that makes me call things even money with no information, can cause real trouble in the real world. How does this problem show that?

jason1990

06-04-2007, 12:22 PM

[ QUOTE ]
Thanks for the reply. Will you also be replying to this:

[ QUOTE ]
Am I correct in assuming your response is: "There is no answer to question (a). THE probability of heads for your bent coin does not exist. Same for (b)."

[/ QUOTE ]

[/ QUOTE ]
I guess not. David, I have now asked you this question four times: here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10635552&page=0&vc=1 ), here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10638568&page=0&vc=1 ), here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10639650&page=0&vc=1 ), and here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10641486&page=0&vc=1 ). Why do you refuse to answer? Do you think it is a trick question designed to make you look stupid? Are you not confident in your opinions on this subject? Are you waiting for Persi to tell you the answer? I, and probably others here, are interested in your opinion on this subject. You are the one who started two threads debating it, so you obviously have a strong opinion about it. But it does not matter if you ignore the question. You have essentially answered it when you said "My point is that there is no such thing as THE probability of an event." You have practically quoted de Finetti, who made the famous statement "PROBABILITY DOES NOT EXIST." De Finetti was essentially the founder of subjective probability. You, David, are clearly a Bayesian.

A lot of posters do not understand these threads, as evidenced by silly comments about asteroids hitting the Earth and so on. David claims to not understand what I am talking about. So here is one last attempt to clarify the debate. Others can clarify beyond this. Maybe even Persi Diaconis will join in the fray.

The debate centers on my claim that David is being inconsistent. On the one hand, he wants to adhere to Bayesianism and claim that "there is no such thing as THE probability of an event." On the other hand, he wants to claim that his probabilities are not only objective, but in fact supported by science, experiment, and empirical evidence. He might have Phil153 on his side, who said (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10646628&page=0&vc=1 )

[ QUOTE ]
Our "best probability estimate" is subjective? I have to strongly disagree. If we're using the known information in a rational way to arrive at a number, it is not at all subjective. Possibly inaccurate in the light of further information - but not subjective.

[/ QUOTE ]
Phil and David learn about the bent penny on my desk and "guess" that the probability of heads is 0.5. This is based on their total lack of information. They want to say that this "guess" is not subjective. The word "subjective" is a bit ambiguous, but there are two senses of the word that apply to their guess.

First, their guess is relative to an observer. They have no information, but I have a lot of information. I have been flipping the penny ever since I bent it the other night. So my "guess" is different. I gave my wife a partial list of the flip results, but not all of them. So her guess is different from both mine and David's. If these guesses depend on the guesser (or on the guesser's information), then they do not represent any objective reality. In this sense of the word, they are subjective.

Second, their guess is not falsifiable. Suppose chezlaw, who also has no information, says that his best guess is 0.4. David and Phil will obviously claim that their guess is better than his. But no experiment, no amount of empirical evidence, will ever prove them right. Even flipping the coin will not help. They are Bayesians. Their guess will change when they flip the coin. Yet they will still claim that before they flipped the coin, their guess was better. The only argument they have for this is philosophical. Granted, it may be convincing philosophy. Also, they are in good company when they advocate their philosophical arguments. For example, from http://en.wikipedia.org/wiki/Bayesian_probability :

[ QUOTE ]
Advocates of logical (or objective epistemic) probability, such as Harold Jeffreys, Rudolf Carnap, Richard Threlkeld Cox and Edwin Jaynes, hope to codify techniques whereby any two persons having the same information relevant to the truth of an uncertain proposition would calculate the same probability. Such probabilities are not relative to the person but to the epistemic situation, and thus lie somewhere between subjective and objective. However, the methods proposed are controversial. Critics challenge the claim that there are grounds for preferring one degree of belief over another in the absence of information about the facts to which those beliefs refer. Another problem is that the techniques developed so far are inadequate for dealing with realistic cases.

[/ QUOTE ]
However, it is still philosophy. Their claims are not falsifiable through experiment and empirical evidence. In this sense of the word, their guess is still subjective.

De Finetti held the view that objective probabilities do not exist. That is, all probability statements are non-falsifiable. In de Finetti's philosophy, there is no falsifiable distinction between David's guess and chezlaw's guess. From the point of view of falsifiability, neither holds any special position. Even Persi Diaconis will tell you that.

So if David and Phil (and others) want to make subjective guesses, then they are free to do that. If they want to argue, on philosophical grounds, that their guess is better than anyone else's who might disagree, then they will find a camp of very smart philosophers who will agree with them. But David wants to go a step farther than that. He wants to imply that his position is supported by science, experiment, and empirical evidence. It is not.

wtfsvi

06-04-2007, 12:31 PM

I agree that the statement in relation to the bent coin "there is 50% chance of heads" is more of a philosophical statement than a scientific one. So I guess I agree with you afterall, Jason.

David Sklansky

06-04-2007, 12:40 PM

Second, their guess is not falsifiable. Suppose chezlaw, who also has no information, says that his best guess is 0.4. David and Phil will obviously claim that their guess is better than his. But no experiment, no amount of empirical evidence, will ever prove them right.

Logic doesn't prove us right?

As far as empirical evidence, what about the past monetary history of people who have taken less than even money when they have no information.

High Falootin

06-04-2007, 12:58 PM

yo guys I think this is pretty simple. If you want objective truth, it only exists subjectively. If you're looking for subjective truth, it can exist objectively.

But objective truth can not exist objectively. But then again, who cares? Subjectivity is boss.

Now let's all chill out and be friends.

jason1990

06-04-2007, 01:08 PM

djames

06-04-2007, 01:20 PM

[ QUOTE ]
"Why do you consistently question obvious intelligence by interrogating for educational background while namedropping when you yourself must hide behind an "I was too lazy" copout to explain why you don't have the credentials you require from posters that disagree with you?"

If someone is disagreeing with me about something that I am almost sure about regarding mathematically related ideas, they are almost always wrong. If they back their disagreement up with techical arguments and they are obviously well educated and intelligent, I'll move them up to a small favorite over me. Unless they are extremely intelligent or well educated. Then I concede and perhaps study what they are saying. And since I actually know Persi Diaconis, I might be able to persuade him to look at these threads and post. If I get lucky he will agree with me and I can avoid some work.

[/ QUOTE ]

Superb. I look forward to his responses. I'm betting that no flaws with be uncovered in the posts made by jason1990. I have uncovered none, and neither have you. I also pretty much ignore the ravings made by many in these threads, but not due to the arrogance that you've displayed on countless occasions. I for one am not as conceited as you clearly are.

PairTheBoard

06-04-2007, 01:45 PM

[ QUOTE ]
Second, their guess is not falsifiable. Suppose chezlaw, who also has no information, says that his best guess is 0.4. David and Phil will obviously claim that their guess is better than his. But no experiment, no amount of empirical evidence, will ever prove them right.

Logic doesn't prove us right?

As far as empirical evidence, what about the past monetary history of people who have taken less than even money when they have no information.

[/ QUOTE ]

What Logic? What proof? Define "no information". As soon as you define the two Choices you have the information about them that defines them. The differences in opinion about the "probability" of the two choices arise from the inferences people draw about that information. People don't bet on A or B. You have to tell them what A and B are before they will bet on them. Once you tell them what A and B are, there is information availbable. How can you be sure that the information contained in the definition of the two choices provides no implications about which choice to prefer?

How are you going to set up your empirical evidence gathering? Are you going to be the one deciding when there are no implications for preference from the infomation contained in the definition of the two choices? In the bent coin case you just proclaim there are no such implications for preference between Heads and Tails. Others might think they do see implications for preference. In your empirical study are you also going to make such proclamations? In my Two Envelope example people also want to make such a proclamation. But it turns out there really are implications stemming from the definition of the two choices.

What you are saying is that you can find an empirical record of money bet on choices where people disagreed on the implications of the definition of the two choices and where the Sklansky proclamation is that there are no preferential implications. You will then compute the success rate of your Sklansky proclamations. But all this empirical study will prove is that Sklansky is a better gambler than most people.

I'd like to see this empirical study. Can I see it? Is there a subset of the data which shows the Sklansky Proclamations did not always prove so statistically successful? Do we have to call up David Sklansky every time we want to determine when we have a "no information" situation where the information contained in the defintion of the two choices has no preferential implications? Or should we call Persi Diaconis instead?

PairTheBoard

Phil153

06-04-2007, 02:04 PM

[ QUOTE ]
De Finetti held the view that objective probabilities do not exist. That is, all probability statements are non-falsifiable. In de Finetti's philosophy, there is no falsifiable distinction between David's guess and chezlaw's guess.

[/ QUOTE ]
This is an extremely strong statement, that has to be wrong. We can test the validity of a probability estimate by repeating trials, and gain a confidence interval for its correctness (or a positive balance in our poker account).

For example, I can say:

Statement 1: "When rolling an unknown 6 sided dice, with no other information about the dice, the most probable distribution of rolls is one where each number has an equal (1/6) chance of coming up".

In the absence of other information, I would argue that this is vastly superior to:

Statement 2: "When rolling an unknown 6 sided dice, with no other information about the dice, the most probable distribution of rolls is one where 4 comes up 95% of the time, and the other five come up 1% of the time"

Do you disagree? Are you telling me these statements are objectively equivalent and equally likely to be correct? The dice could be just as loaded as your coin in this particular case. Does that make statement 1's correctness equal to statement 2's?

Perhaps there's some deep point you're making that i don't see. I don't think you are based on a couple of comments.

SamIAm

06-04-2007, 02:11 PM

[ QUOTE ]
I'm not sure if the paradox relates to anything I've said or not. But even if it does, it is a contrived example involving numbers approaching infinity and PTB is implying that the indifference principle, or whatever it is that makes me call things even money with no information, can cause real trouble in the real world. How does this problem show that?

[/ QUOTE ]
I would say that the Envelope Paradox comes about by assuming that because we don't know the distribution, all numbers are equally likely.

djames

06-04-2007, 02:19 PM

Phil, this
[ QUOTE ]
[ QUOTE ]
De Finetti held the view that objective probabilities do not exist. That is, all probability statements are non-falsifiable. In de Finetti's philosophy, there is no falsifiable distinction between David's guess and chezlaw's guess.

[/ QUOTE ]
This is an extremely strong statement, that has to be wrong.

[/ QUOTE ]
and this
[ QUOTE ]
Perhaps there's some deep point you're making that i don't see. I don't think you are based on a couple of comments.

[/ QUOTE ]
has me confused. Are you saying that De Finetti's position "has to be wrong" with your dice argument that follows or that Jason has to be wrong? Please take care in reading the post that you quoted, which states the opinion held by De Finetti. This first part of your latest post actually reads like you're arguing with Jason and against De Finetti (and thus Sklansky) while the tone of your posts suggests you think you're doing the opposite, at least from my read.

Based on this confusion, I will admit that I did not give your two dice examples any thought whatsoever. I don't know whether those examples argue against De Finetti, Jason or Sklansky at this point.

Phil153

06-04-2007, 02:22 PM

Against Finetti, and by extension, against Jason. Admittedly I don't know how Jason's point argues anything other than Finetti. Perhaps I'm not smart enough, I do not know.

jason1990

06-04-2007, 03:00 PM

[ QUOTE ]
Against Finetti, and by extension, against Jason. Admittedly I don't know how Jason's point argues anything other than Finetti. Perhaps I'm not smart enough, I do not know.

[/ QUOTE ]
You and David are essentially adopting the position of logical, or objective epistemic, probability. This is generally regarded as a variety of Bayesian probability. De Finetti's work lies at the foundations of Bayesian probability. I tell you this so you know the history of your own philosophy.

There is a controversy surrounding your point of view. Brilliant thinkers are arguing on both sides of this controversy.

I am not advocating either side. I am simply pointing out that your side does not presently have a scientific proof that you are correct. David thinks he does. He does not.

Phil153

06-04-2007, 03:12 PM

I know nothing of Finetti except from your quotes. My major was physics with only a small interest in math. However, I disagree with the position below that you attribute to Finetti, for reasons stated in my post above.

[ QUOTE ]
De Finetti held the view that objective probabilities do not exist. That is, all probability statements are non-falsifiable. In de Finetti's philosophy, there is no falsifiable distinction between David's guess and chezlaw's guess. From the point of view of falsifiability, neither holds any special position.

[/ QUOTE ]

I consider it obvious that this is wrong. I'm pretty sure David does too. So I don't understand how you think I hold the same position as De Finetti...

Anyway, thanks for the debate. This obviously requires further reading on my part.

f97tosc

06-04-2007, 03:40 PM

On bent and even coins.

In this post I will try to write down some of my thoughts on problems on the bent and fair coins. The discussion seems to be mixed up over multiple threads and I don't think I have read everything, so my apologies if I address something already covered or if my post is misaligned with original question, which I haven’t found. With those disclaimers out of the way, here goes.

First of all, I think in these fissiparous debates over the nature of probability, it is useful to bring the attitude that a probability is simply something we define; there may be several different definitions that are perfectly rigorous, and different definitions may be more or less useful for specific problems.

I think we can all agree that a probability definition based on frequency, or even on the physical properties of the bent coin, is not useful for making statements about the first toss of a bent coin with unknown or incompletely specified properties. In that case, the probability and any associated wagering problem are simply ill-defined.

So the question is then, is there some other definition of probability that is not only rigorous, but also useful for anybody trying to make decisions involving bent coins, but lacking a large number of observations? Is a Bayesian definition, based on symmetry of information, up to the job? I don't think anybody disputes Bayes' theorem, so the question is just if, for a given problem statement, there is a rigorous way to represent the initial information in terms of prior probabilities. If the initial problem says nothing that hints that heads or tails will be more likely, then that is as rigorous specification as any that we have no information favoring one over the other; we can mathematically state this as P=0.5 for the first toss of the bent coin (just the same as for the fair coin). Any wagering decision based on this assignment may not be optimal in the eyes of somebody who knows the properties of the bent coin, but it will be the best we can do based on the information we have.

For the first toss, the situation with the bent coin is in fact almost exactly analogous to the fair coin. In the latter case, we are ignorant of the exact nature of the "random" coin-tossing procedure, but there is a certain symmetry of information saying that we can’t tell which one is more likely to come out. Of course, in reality, both the bent and the fair coin will either turn up heads or tails following deterministic laws of physics (I am assuming no quantum effects here), it is just that we don't know the exact trajectory of the coin or its exact initial placement. Some other person may know more about the bent coin and be able to say more about likely outcomes, just like a trained observer may be able to predict the outcome of a specific toss of a fair coin with greater than 50% probability. But for the problem at hand we are assuming that we are not that knowledgeable person.

Thus when tossing a fair or a bent coin for the first time, it may be that the decision-maker, because of his or her lack of information (and not because of any physical non-determinism) feels perfectly ignorant of the outcome of the toss. We then encode this ignorance with P=0.5. It is indisputable that any wagering decisions derived from here, using Bayes’ theorem and common arithmetic, are logically consistent with each other, as well as with the decision-maker's initial knowledge about the coin, as specified in the problem formulation (assuming that the specification indeed was symmetric with respect to heads and tails). I don't know what else one could possibly ask for from a mathematical theory for decision-making under imperfect information.

Suppose now that the bent coin is tossed once and we learn the result, and we would like to make statements (or wagers) about subsequent tosses. Now things get more complex. Suppose that we write f(P) for the prior probability distribution that the coin is such that when tossed, it will result in heads a long-term fraction P of the time. Clearly, if we have f(P) specified everything we may want to know about wagering after a certain set of observations (including zero or one tosses) will be known thru Bayes theorem. But is there enough information in the problem to rigorously specify f(P)? Well I haven't found the original problem but there may well not be. It is easy to specify perfect ignorance with respect to two discrete alternatives (P=0.5), but it is much trickier with respect to a continuous space representing the possible curvatures of the coin. And in a real application we may well not be perfectly ignorant; common knowledge about how coins may be bent may suggest that some values of P are more likely than others. There is a great deal of esoteric math involving stuff like transformation groups that could help us go from various statements about the coin into specific instances of f(P) (I think I cited a well-known paper in another thread). But in order to apply these tools one often needs a more well-specified problem than is common in applications.

On the other hand, even if the problem formulation is not clear enough to completely specify f(P), there may well be enough information to answer some questions of interests, or to give us useful structural insights about the problem. For example, if we again assume that we are perfectly ignorant about the outcome of the first toss, that already places certain constraints on f(P). From there, we only need to make some very mild additional technical assumptions to be able to conclude that, for example, after seeing one head, we rationally should judge it more likely than not for the next toss to be head as well (details on this available upon request).

PairTheBoard

06-04-2007, 03:43 PM

You have two choices, A and B. You must define them. In that definition lies information. Suppose you claim there are no logical inferences for prefering one over the other. At what point do you stop looking for logical inferences? In the Two Envelope example, many people stop looking too soon. In fact, even after lengthy explanations for why they should consider inferences contained in the defintion of the Choices, they still maintain their original position looks right and they should be able to compute expectations from the subjective probabilities of 50-50 they perceive.

So in general, when do you stop looking for the logical inferences. In the bent coin example, I can see somebody very reasonably doing something like this. They ask, how are coins likely to be bent by someone? If I were to bend a coin how would I do it? I would probably hold the coin in front of me Heads Up. That's usually the way I look at a coin. I think of Heads as the front of the coin. I would then probably grab the sides of the coin and push my thumbs in on the middle. Thus making a larger surface area for a flip to land Tails-Down rather than Heads-Down. I might then estimate this to be a general bias for people who bend coins and estimate a measure of preference slightly in favor of Heads-Up for the coin on Jason's desk.

So at what point do we stop looking for inferences from the information contained in the definition of the two Choices? Do you just proclaim no such inferences exist? Do you object that these are not logical inferences but ones that bring in additional information? If so, how do you know you have examined all the purely logical inferences of the Definitional Information. In the Two Envelope example the inferences are purely mathematical. Yet people overlook them because they stop looking for inferences too soon. How do we really apply this principle of yours in practice? And if we tried to test it empirically, how would you define the test cases?

PairTheBoard

Beavis68

06-04-2007, 03:48 PM

I am not sure if anyone has done the math, but I think this is it.

Chance of the bent coin coming up heads = x chance of the bent coin coming up tails = 1-x.

Chance of you flipping heads with your coin when the bent coin is heads = 1/2 odds of flipping tails when the bent coin comes up tails = 1/2

The probability of the bent coin being heads and you flipping heads is x*1/2=1/2x

Probability of the bent coin coming up tails and you fillping tails on your coin = (1-x)*1/2 = 1/2 - 1/2x.

the probablility you will flip the correctly and win = 1/2x + (1/2 - 1/2x) = 1/2

What I got my brain around that made me accept this is that when you flip whatever why the coin is baised too, you will be correct much more often, so this even it out.

the bias (x) is not relavant to the equation

TomCowley

06-04-2007, 04:10 PM

I fail to see why this isn't falsifiable in a gambling game. There exists a coin with P(heads) of X and P(tails) of 1-X. You have no knowledge of which side is favored, nor of how much it is favored by.

Somebody tells you that a person with perfect knowledge of the coin will make one flip and call it. You are playing a gambling game, and are required to pick a number to represent P(h). The game pays out as follows. If the person calls heads, he wins 1-Ph if he wins the flip and loses P(h) if he loses the flip. Same for tails, except he wins P(h) and loses 1-P(h).

If P(h) is the actual probability of heads, this is a fair game. If your estimated p(h) is too low, then the person has an edge by calling heads and getting a disproportionately big payout. If your estimated p(h) is too high, then the person has an edge by calling tails.

Clearly, the optimal strategy for this game is to pick the number that minimizes your disadvantage over all possible coins. The edge (for P(h) too low) is realP(h)*(1-P(h))-(1-realP(h))*P(h)=rph-rph*ph-ph+rph*ph=rph-ph, and it's obvious that the edge is |rph-ph|.

Therefore the EV of the game is Integral x from 0-1 of |x-ph|.

=Integral 0 to ph of ph-x + Integral ph to 1 of x-ph
= phx -x^2/2 ]ph,0 + x^2/2-xph]1,ph
=ph^2 -ph^2/2 + 1/2 -ph -ph^2/2 +ph^2
= ph^2 -ph +1/2

Now the best strategy minimizes this function, so take the derivative and look for the zeros, and you get 2ph-1=0, ph=0.5

It is therefore demonstrable that the BEST guess, given no knowledge, is 0.5. Refusing to state this as the odds line is just as stupid as refusing to state odds on a roulette wheel "out of a desire to remain objective".

About such people, you said:

[ QUOTE ]
But if met someone who refused to assign a numeric value to the probability that I win my next bet at the roulette wheel, and justified it by claiming a desire to remain as objective as possible, then I would consider that person to be in denial about the practical reality of probability.

[/ QUOTE ]

Refusing to place the odds line on the coin at 0.5 is denying the practical reality of probability, and any objection based on that is worthless. Of course we never know what P(h) actually is, but the DEMONSTRABLE best guess is 0.5.

SamIAm

06-04-2007, 05:30 PM

[ QUOTE ]
Clearly, the optimal strategy for this game is to pick the number that minimizes your disadvantage over all possible coins.

[/ QUOTE ]
I don't think it's so clear. You're assuming that all possible coins are equally likely. (Also that our utility function is linear wrt the distance from the true value. 50% is almost definitely not the right guess; it's just the guess that won't be very right or very wrong.)

[ QUOTE ]
I fail to see why this isn't falsifiable in a gambling game.

[/ QUOTE ]
I think one of the points of this thread is that you can't recreate the conditions of the problem in the real world. If we use statements like "somebody bends a coin and we have no knowledge about how they bend it", then as soon as we pick somebody to actually bend a coin, we gain knowledge. There are internally consistent arguments that hold up well, but that classifies it as philosophy instead of math.

Pokerlogist

06-04-2007, 08:02 PM

If the coin was known to be bent in a random direction, ie in an unbiased way, then that would be enough information to say that .50 is a reasonable estimate of the probability of heads being flipped.

shday

06-04-2007, 08:16 PM

The fairness of the coin is irrelevant. The chance of winning will always be 50%. The sensible bet is on the bent coin.

SamIAm

06-04-2007, 09:17 PM

[ QUOTE ]
If the coin was known to be bent in a random direction, ie in an unbiased way, then that would be enough information to say that .50 is a reasonable estimate of the probability of heads being flipped.

[/ QUOTE ]
While true, this isn't what we're talking about. There's no knowledge of randomness. There's no knowledge at all.

PairTheBoard

06-04-2007, 09:29 PM

[ QUOTE ]
If the coin was known to be bent in a random direction, ie in an unbiased way, then that would be enough information to say that .50 is a reasonable estimate of the probability of heads being flipped.

[/ QUOTE ]

That's almost true by definition if you qualify "random" with "unbiased random". It's also something you could test under controlled conditions. Maybe produce a bunch of unbiased randomly bent coins by flipping fair coins and inserting them in a bending machine. The machine could even vary the bend as long as you inserted the coins randomly face up. You could then flip a bunch of these coins and record the frequency of Heads.

In reality the question is how confident you are that the bending was indeed random and unbiased. Just like with the Cards, how confident are you that you have a well shuffled deck. Was it shuffled 7 times like it should be? Is even a 7 shuffled deck precisely random?

In the Two Envelope problem, people are sometimes told the envelope amounts were chosen randomly. What they don't realize is that it's impossible that such a random choosing of envelope amounts is an unbiased one in the sense that all envelope amounts are equally likely.

It's been argued that there is an obvious symmetry to the Heads, Tails in the coin situation which allows us to "logically" conclude equal likelihood for Heads as Tails. There may be approximate symmetry in the situation, but as soon as you identify the two choices you have automatically broken pure symmetry. For pure symmetry with the coin you would need one side of the coin to be indistinguishable from the other. You might then logically argue that by symmetry both sides are equally likely to come up. The trouble is, you wouldn't be able to tell because the two sides would be indistinguishable. Once you identify Head from Tails you have two different physical features to the two sides of the coin. You have lost pure logical symmetry.

This is why Sklansky's statement about all propositions A,B having no other information about them, historically being equally likely, holds no information for us. Which was equally likely with the other? To test this assertion you would have to consider a classs of such pairs and show that you can't find a bias by picking A over B. Which A though? Which B? You can't tell until you actually identify A and B and pick one out according to some criteria. Once you do that, symmetry is broken. You are merely doing a test on the class you have defined and the criteria you've chosen for picking A or B. If your criteria is to flip a coin to make your pick, then your are right back into the trivial observation that flipping a coin gives you a 50-50 chance of making the right pick. It says nothing about P(A) or P(B).

PairTheBoard

Beavis68

06-04-2007, 10:29 PM

there is a 50% chance it is biased towards heads and there is a 50% chance it is biased to tails.

But this has nothing to do with the original problem as I understand it.

When you guess if it is heads or tails, regardless of the bias you have a 50/50 chance of being correct

PairTheBoard

06-04-2007, 10:51 PM

[ QUOTE ]
there is a 50% chance it is biased towards heads and there is a 50% chance it is biased to tails.

[/ QUOTE ]

Not necessarily. Suppose you engaged in the following analysis:
------------------
So in general, when do you stop looking for the logical inferences. In the bent coin example, I can see somebody very reasonably doing something like this. They ask, how are coins likely to be bent by someone? If I were to bend a coin how would I do it? I would probably hold the coin in front of me Heads Up. That's usually the way I look at a coin. I think of Heads as the front of the coin. I would then probably grab the sides of the coin and push my thumbs in on the middle. Thus making a larger surface area for a flip to land Tails-Down rather than Heads-Down. I might then estimate this to be a general bias for people who bend coins and estimate a measure of preference slightly in favor of Heads-Up for the coin on Jason's desk.
----------------------

[ QUOTE ]

But this has nothing to do with the original problem as I understand it.

When you guess if it is heads or tails, regardless of the bias you have a 50/50 chance of being correct

[/ QUOTE ]

If you flip a fair coin to make your guess, yes. But altough he keeps dodging the question, Sklansky continues to imply that this says something about specifically P(Heads) and P(Tails) for the bent coin. It doesn't mathematically. He can philosophically make that assertion. But applying it as a general principle for computing more probabilites and EV's has its pitfalls. You can take two obviously non-equally likely events like Rolling Snake Eyes or Not Rolling Snake Eyes, flip a fair coin to make your guess for which will happen and be assured you will guess right half the time. That hardly implies that P(Snake Eyes) is 50%.

PairTheBoard

TomCowley

06-04-2007, 11:15 PM

The heads-tails symmetry isn't necessary to get a guess of 50%. Say you're rolling a weighted 6-sided die and trying to predict P(1). It can be weighted in any way, in any degree, so P(1) can be anywhere from 0 to 1. The best guess is still 50%.

Saying "no other information is available beyond the possible outcomes (happens or doesn't happen)" is best interpreted as a uniform prior distribution. You can't possibly know one outcome (of repeated trials with the same coin) is more likely than any other (which is true if P is anything but uniform) without any information. From that, when trying to set a money line, 50% is the only reasonable number to choose. P(heads) in any one trial is almost surely not 50%, but if you must use a number, it is demonstrably the best number to choose against sharp bettors.

No other argument provides a number to use. I could give a flying rat's ass about philosophy. If somebody puts a gun to my head and forces me to make a moneyline for my own money on a happens/doesn't happen that I have no information about beyond those two possible outcomes, I'd be a retard to choose any other number. So would the rest of you, and you all know it. Arguing anything beyond that has no practical relevance. (and 2 envelopes is not relevant to a happens/doesn't happen prediction)

To prove that you have a RELEVANT point, demonstrate a practical happens/doesn't happen example, with no information beyond the outcomes, where the proper money line to choose is not 50%.

PLOlover

06-04-2007, 11:15 PM

[ QUOTE ]
You can take two obviously non-equally likely events like Rolling Snake Eyes or Not Rolling Snake Eyes, flip a fair coin to make your guess for which will happen and be assured you will guess right half the time. That hardly implies that P(Snake Eyes) is 50%.

[/ QUOTE ]

well if the dice are prerolled than if you're getting the right odds (35-1 for snake eyes I think) for your choice then it doesnt matter if the dice are loaded or not.

Since there's only one price for a coin flip no matter which side you choose, as long as you flip it first to decide the outcome and then (darkly) make your pick, it won't matter if the coin is bent or not, or even if it is two headed.

TomCowley

06-04-2007, 11:23 PM

[ QUOTE ]
I don't think it's so clear. You're assuming that all possible coins are equally likely.

[/ QUOTE ]

Of course. How could I say any one is more likely than any other without extra knowledge? The point is that if you MUST create a guess, 50% is demonstrably the only rational guess. When "insufficient information" isn't acceptable, and action must be taken, 50% is the only rational action. Arguing beyond that has no practical relevance.

PairTheBoard

06-04-2007, 11:50 PM

[ QUOTE ]
[ QUOTE ]
You can take two obviously non-equally likely events like Rolling Snake Eyes or Not Rolling Snake Eyes, flip a fair coin to make your guess for which will happen and be assured you will guess right half the time. That hardly implies that P(Snake Eyes) is 50%.

[/ QUOTE ]

well if the dice are prerolled than if you're getting the right odds (35-1 for snake eyes I think) for your choice then it doesnt matter if the dice are loaded or not.

Since there's only one price for a coin flip no matter which side you choose, as long as you flip it first to decide the outcome and then (darkly) make your pick, it won't matter if the coin is bent or not, or even if it is two headed.

[/ QUOTE ]

I don't think you understood my post.

PairTheBoard

Phil153

06-05-2007, 01:29 AM

I don't find your arguments compelling. Your eternal doubt that the process can produce anything useful doesn't fit with the fact that in the real world, it does produce useful results.

BTW, f97tosc dropped a H-bomb on this thread. A very smart man.

PLOlover

06-05-2007, 01:35 AM

[ QUOTE ]
Quote:

Quote:
You can take two obviously non-equally likely events like Rolling Snake Eyes or Not Rolling Snake Eyes, flip a fair coin to make your guess for which will happen and be assured you will guess right half the time. That hardly implies that P(Snake Eyes) is 50%.

well if the dice are prerolled than if you're getting the right odds (35-1 for snake eyes I think) for your choice then it doesnt matter if the dice are loaded or not.

Since there's only one price for a coin flip no matter which side you choose, as long as you flip it first to decide the outcome and then (darkly) make your pick, it won't matter if the coin is bent or not, or even if it is two headed.

I don't think you understood my post.

PairTheBoard

[/ QUOTE ]

well why does it matter whether you make your choice after the outcome is known but unknown to you, or whether you make your choice before the outcome is known? If it's 50/50 in one case it must be 50/50 in the other.

PairTheBoard

06-05-2007, 02:40 AM

[ QUOTE ]
I don't find your arguments compelling. Your eternal doubt that the process can produce anything useful doesn't fit with the fact that in the real world, it does produce useful results.

BTW, f97tosc dropped a H-bomb on this thread. A very smart man.

[/ QUOTE ]

What "process" are you talking about? If you are talking about the process by which Baysian Statisticians assume Prior distributions often based on the indifference principle, then you are wrong. I realize this process can often be very useful. It can also be misleading if it's not made clear that the conclusions are based on not just the data but also the assumed prior distribution.

As far as f97tosc's H-bomb, its nucleus is the following:

[ QUOTE ]
If the initial problem says nothing that hints that heads or tails will be more likely, then that is as rigorous specification as any that we have no information favoring one over the other; we can mathematically state this as P=0.5 for the first toss of the bent coin

[/ QUOTE ]

What's in bold is what people can disagree on. In the Bent Coin example it looks like a safe assumption. But I gave an example above - which appears in two of my posts - of some plausible "hints" that might be inferred about the way the coin may have been Bent. Did you read that example or should I copy it again? I don't see you having responded directly to it or to most of my other observations. How do we apply this principle in general? How do we determine that there truly is no "hint" to one choice being more likely than the other? Do we just take your word for it? Or must we call Sklansky each time to see what he says? Or maybe we should call Persi Diaconis.

PairTheBoard

djames

06-05-2007, 10:59 AM

[ QUOTE ]
Saying "no other information is available beyond the possible outcomes (happens or doesn't happen)" is best interpreted as a uniform prior distribution. You can't possibly know one outcome (of repeated trials with the same coin) is more likely than any other (which is true if P is anything but uniform) without any information. From that, when trying to set a money line, 50% is the only reasonable number to choose. P(heads) in any one trial is almost surely not 50%, but if you must use a number, it is demonstrably the best number to choose against sharp bettors.

No other argument provides a number to use. I could give a flying rat's ass about philosophy. If somebody puts a gun to my head and forces me to make a moneyline for my own money on a happens/doesn't happen that I have no information about beyond those two possible outcomes, I'd be a retard to choose any other number. So would the rest of you, and you all know it.

[/ QUOTE ]

Uhh.... A gun is pointed at your head. You will see a perfectly fair coin tossed in front of you. On one side of the coin is the number one. On the other is the number zero. You have a chance to predict the next outcome of a toss of this coin. If you are wrong, you will die. If you are right, you will live on. You're picking 0.5? Nice knowing ya!

luckyme

06-05-2007, 11:05 AM

[ QUOTE ]
How do we determine that there truly is no "hint" to one choice being more likely than the other?

[/ QUOTE ]

Is there a difference between a hint we don't notice and a hint that wasn't issued?

luckyme

PairTheBoard

06-05-2007, 12:00 PM

[ QUOTE ]
[ QUOTE ]
How do we determine that there truly is no "hint" to one choice being more likely than the other?

[/ QUOTE ]

Is there a difference between a hint we don't notice and a hint that wasn't issued?

luckyme

[/ QUOTE ]

There could be a very big difference. Why didn't we notice it? Did we fail to notice because we were so confident in applying the Indifference Priciple that we failed to look for it? That's what happens with people in the Two Envelope problem. They are convinced the Indifference Principle applies. So they fail to look at the mathematical Hints in the problem. The same thing happens to people in the Monty Hall problem. They are so convinced the Indifference Principle applies that they sometimes fail to take the opportunity to switch doors.

I'm not saying it's not useful to sometimes apply the Indifference Principle. If we were to apply the Baysian method to model a probability distribution for the real P(Heads), say p, by starting with a Prior Distribution, flipping the coin, then recomputing the Distribution based on the outcome of the flip, most people would agree that applying the indifference principle and starting with a uniform Prior Distribution for p is the way to go. However, even then the results are controversial and the Uniform Prior Distribution is subjective.

Sometimes a Prior Distribution can be guessed at which purports to contain prior information. There are strong arguments for the usefulness in that case when compared to conventional statistical methods which only look at the data produced by flipping the coin, and don't try to model a probability distribution for p itself.

Suppose we look at P(Next Dice Roll) for honest dice. We could also say there might be Hints we haven't looked for that could produce bias. We don't bother looking for them every time we roll the dice do we? And it's very useful for us not to bother. We do quite well not bothering and making our bets based on the assumption that there is no bias. Why is that different? The reason is because we have a great deal of experience with the situation. It has proved a good working assumption by way of lengthy testing.

But even then we didn't make that assumption lightly. We may want to test the dice for balance just to make sure. However, I'm pretty certain that if you were gambling with those dice in a strange game and asked to test them on your Balancing Tool and someone put a Gun to your head saying, Don't Bother keep gambling, you would think twice about your original assumption that there is no bias to the dice.

That is essentially what I am objecting to on these kinds of things. I do not want someone putting a gun to my head forcing me to stop looking for what Hints might lie in the problem settup by proclaiming we must apply the Indifference Principle. In the case of rolling dice in a Casino where I am confident they are aproximately unbiased, I have no problem assuming they are unbiased for my gambling decisions. I am familiar with that situation. But when the situation is a strange one where the preconditions are vaugely defined and the Indifference Principle is not one I can test, I am going to be very suspicious when applications of it produce counter intuitive results.

PairTheBoard

TomCowley

06-05-2007, 12:02 PM

[ QUOTE ]
Uhh.... A gun is pointed at your head. You will see a perfectly fair coin tossed in front of you. On one side of the coin is the number one. On the other is the number zero. You have a chance to predict the next outcome of a toss of this coin. If you are wrong, you will die. If you are right, you will live on. You're picking 0.5? Nice knowing ya!

[/ QUOTE ]

Try reading the post before making inane comments. You're so far off that it actually is funny.

djames

06-05-2007, 12:58 PM

[ QUOTE ]
[ QUOTE ]
Saying "no other information is available beyond the possible outcomes (happens or doesn't happen)" is best interpreted as a uniform prior distribution. You can't possibly know one outcome (of repeated trials with the same coin) is more likely than any other (which is true if P is anything but uniform) without any information. From that, when trying to set a money line, 50% is the only reasonable number to choose. P(heads) in any one trial is almost surely not 50%, but if you must use a number, it is demonstrably the best number to choose against sharp bettors.

No other argument provides a number to use. I could give a flying rat's ass about philosophy. If somebody puts a gun to my head and forces me to make a moneyline for my own money on a happens/doesn't happen that I have no information about beyond those two possible outcomes, I'd be a retard to choose any other number. So would the rest of you, and you all know it.

[/ QUOTE ]

Uhh.... A gun is pointed at your head. You will see a perfectly fair coin tossed in front of you. On one side of the coin is the number one. On the other is the number zero. You have a chance to predict the next outcome of a toss of this coin. If you are wrong, you will die. If you are right, you will live on. You're picking 0.5? Nice knowing ya!

[/ QUOTE ]

[ QUOTE ]
Try reading the post before making inane comments. You're so far off that it actually is funny.

[/ QUOTE ]

Oh, I read it, and I didn't misinterpret it at all. I just disagree that choosing the uniform prior isn't a slam dunk. So when you said:

"P(heads) in any one trial is almost surely not 50%, but if you must use a number, it is demonstrably the best number to choose against sharp bettors"

I disagree. Choose 1 or choose 0, but don't choose 0.5. You'll be wrong less than all the time and will thus sometimes survive my hypothetical game of death/life. What is the value in minimizing the "error" of your first choice? Why not minimize the frequency your first choice is incorrect? (Please note that if the bet is "truly" 50/50, the expected absolute errors of the first flip of a prior of 0.5 and a prior of 0/1 are equal anyway.)

So when you go on to say:
"If somebody puts a gun to my head and forces me to make a moneyline for my own money on a happens/doesn't happen that I have no information about beyond those two possible outcomes, I'd be a retard to choose any other number. So would the rest of you, and you all know it,"

where the number you'll choose was stated to be 0.5 in the prior paragraph, and then follow up that goodness with:

"Try reading the post before making inane comments. You're so far off that it actually is funny,"

it is I that gets to laugh, since you'll be dead 100% of the time.

TomCowley

06-05-2007, 01:35 PM

[ QUOTE ]
"P(heads) in any one trial is almost surely not 50%, but if you must use a number, it is demonstrably the best number to choose against sharp bettors"

I disagree. Choose 1 or choose 0, but don't choose 0.5. You'll be wrong less than all the time and will thus sometimes survive my hypothetical game of death/life. What is the value in minimizing the "error" of your first choice? Why not minimize the frequency your first choice is incorrect? (Please note that if the bet is "truly" 50/50, the expected absolute errors of the first flip of a prior of 0.5 and a prior of 0/1 are equal anyway.)

[/ QUOTE ]

I'm talking about the money line to lay on the outcome, you're suggesting that I was actually SUGGESTING that the outcome is 0.5, or that making a choice based on an estimated probability of 0.5 would not result in a guess of 0 or 1. WTF? Next.

djames

06-05-2007, 02:33 PM

You responded to a post by PTB, who tried to get you back on track to the interesting problem.

PTB:
"But altough he keeps dodging the question, Sklansky continues to imply that this says something about specifically P(Heads) and P(Tails) for the bent coin. It doesn't mathematically. He can philosophically make that assertion. But applying it as a general principle for computing more probabilites and EV's has its pitfalls. You can take two obviously non-equally likely events like Rolling Snake Eyes or Not Rolling Snake Eyes, flip a fair coin to make your guess for which will happen and be assured you will guess right half the time. That hardly implies that P(Snake Eyes) is 50%."

Then your post comes with the dice example and the paragraphs I've quoted again.

So, do you care to speak to the problem we care about, or are you content talk about this money line strategy that no one has argued against? I guess I misinterpreted your statements as offering your opinion on the problem we care about, that has been ignored by Sklansky ... probably because you replied to PTB directly you pointed this out to you as well.

PairTheBoard

06-05-2007, 03:28 PM

I have a hard time responding to TC because I've had him on ignore since he told me to STFU and F-off. Unfortunately, I can't help but notice what he says when people quote him. I've followed a bit of your discussion with him. I don't think he understands what you're saying. He's pretty much a one note propagandist, imo, who doesn't respond to anything but himself.

PairTheBoard

TomCowley

06-05-2007, 04:00 PM

I don't care about debating what P(heads) actually is for that particular bent coin, with no information beyond the fact that it is a bent coin. Until I gain more information about that coin, it's pure speculation, and it's an utterly worthless pursuit because it allows me to make no better decisions. Once I know anything else, the original condition no longer applies.

OTOH, if I am required to make a decision that's contingent on P(heads), 0.5 is clearly the correct estimate to use.

This whole debate seems like:

1: You have no way to know P(heads), stop saying you know it.

2: I have to use a number, and I know 0.5 is the right number to use.

1: You have no way to know that number is correct for P(heads)

2: I HAVE TO use a number, 0.5 is appropriate.

etc.

People with a philosophical bent object to "P(heads)=0.5" and prefer "the best estimate of P(heads) is 0.5". People with a practical bent think this distinction is meaningless because it doesn't allow for better decisions.

djames

06-05-2007, 04:26 PM

[ QUOTE ]
I have a hard time responding to TC because I've had him on ignore since he told me to STFU and F-off. Unfortunately, I can't help but notice what he says when people quote him. I've followed a bit of your discussion with him. I don't think he understands what you're saying. He's pretty much a one note propagandist, imo, who doesn't respond to anything but himself.

PairTheBoard

[/ QUOTE ]

You are wise.

PLOlover

06-05-2007, 04:50 PM

[ QUOTE ]
In the case of rolling dice in a Casino where I am confident they are aproximately unbiased, I have no problem assuming they are unbiased for my gambling decisions.

[/ QUOTE ]

actually its the casinos who prefer random dice and the customers who prefer biased dice, even if no one is aware or caused the bias.

KipBond

06-05-2007, 09:38 PM

Wow. I finished reading both of the "Bent Coin" threads -- very interesting stuff.

I'm 80% sure that Jason & PTB are 90% likely to be 99% correct. /images/graemlins/laugh.gif

As such, and being fairly ignorant of various probability theories, I have a question for them:

Do probabilities have accuracies? For example, in this scenario, can we say that the P(Heads)=50% has an accuracy of 0%? Since there is no empirical evidence to support the P(Heads)=50% assumption, it's accuracy is 0.

On the other hand, if we had a coin that has been proven to be almost certainly fair, we can say it's P(Heads)=50% has almost 100% accuracy.

-----------

Or, if we expand the Bent Coin question as such:

A fair coin is flipped, and then bent in such a way that guarantees subsequent flips will always be the same as the result of the first flip. The bent coin is then flipped. What is the probability that the bent coin will be heads?

P(Heads) = 50% w/ 100% accuracy.

However:

A coin is bent in such a way that guarantees flips will always be the same (always heads, or always tails). The bent coin is then flipped. What is the probability that the bent coin will be heads?

P(Heads) = 50% w/ 0% accuracy.

This 2nd scenario is essentially the same as original bent coin question, as I understand it.

I'd definitely appreciate it, PairTheBoard and/or Jason, if you could comment on this. Thanks!

<EOT>

===================================
References:
=======================================
Here are 2 really good posts in the other thread for people that may not want to read the entire thing:

http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10633752
http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10635053

PairTheBoard

06-05-2007, 11:53 PM

[ QUOTE ]
if we had a coin that has been proven to be almost certainly fair, we can say it's P(Heads)=50% has almost 100% accuracy.

[/ QUOTE ]

First of all, be aware that when we use the term "flip a fair coin" in discussion we are usually talking about a mathematical abstraction. It's just a shorthand way of saying we make a choice between two options randomly with 50% probability for each. So the parameter p=50% is automatically 100% accurate because it is defined to be so in the mathematical model.

With a physical coin I doubt you could ever know if is 100% fair. There are two different kinds of statistics that can be done to measure how close the coin is to being fair. Conventional methods would record flips and produce confidence intervals. If the coin is close to fair, with more flips you could get tighter and tighter confidence intervals about p=.5 with higher and higher confidence. With Baysian methods you would treat the parameter p like it was a random variable and starting with an assumed prior distribtion for p, like the uniform, compute a sequence of distributions determined by the outcome of each flip, where the sequence theoretically converges on a point mass distribution for p.

This was recently done in the Probability Forum with an interesting discussion. jason did much of the hard calculations. Although the sequence in theory would converge to a point mass, in practice you would not know what point mass it was converging to. The best you would get with any finite sequence of flips is a tight probablity distribution. You might guess its mean for p but that would just be a guess. The mean will shift around with more flips.

If you assumed a continuous distribution for the parameter p amongst an imaginary universe of such "fair" physical coins the probability of any one of them having a precise p=.5 would be 0 probability.

So these are the kinds of statistical tests you can do to measure how close the physical coin is to being perfectly fair. An additional problem in practice is that each physical flip may alter the coin. So you can't really flip the same coin more than once. So after flipping a coin 1000 times and having recorded the results, and having calculated the statistics for the results, you still have the problem that the coin you now propose confidence in according to the statistics is now no longer the same coin as was used to produce the statistics.

About all you can do at the end of all this is assume the coin has not been significantly altered by the flips and that you have been convinced from the statistical evidence that it's close enough to Fair to assume it is Fair for your purposes, like maybe gambling with it.

So even for a so called "fair" physical coin, there is a question as to what P(heads) really is. For the purposes of gambling we assume it is 50% even though we know it almost certainly is not exactly 50%. But even if it has a bias, if we make our gambling decisions randomly the bias will not hurt us.

[ QUOTE ]
Bent Coin Scenario -
Do probabilities have accuracies? For example, in this scenario, can we say that the P(Heads)=50% has an accuracy of 0%? Since there is no empirical evidence to support the P(Heads)=50% assumption, it's accuracy is 0.

[/ QUOTE ]

From the discussion above you can see that even for the so called "fair" physical coin, P(Heads)=50% is almost certainly not true. For the bent coin we would expect statistics for repeated flips to give us high confidence the long run P(Heads) is outside a significant interval around .5. But we don't have repeated flips to work with for the query in the OP. We are not looking at the long run intrinsic P(Heads) for this coin. We are only looking at P(Heads on the first flip). This creates an entirely different problem for us. How are we to mathematically model this proposed probability? Is it a model we can test statistically? Some people have suggested we model it according to some kind of "random" bending process.

[ QUOTE ]
A fair coin is flipped, and then bent in such a way that guarantees subsequent flips will always be the same as the result of the first flip. The bent coin is then flipped. What is the probability that the bent coin will be heads?

[/ QUOTE ]

I think that's what you are doing here. To simplify things you have made the bend extreme and so that flips of the Bent Coin will always match the result from flipping an initial "Fair" Coin. If this Fair Coin is the mathematically abstract one I talked about above, you have now proposed a precise mathematical model with which you Could conclude P(Heads on first flip of Bent Coin) = P(Heads for abstract fair coin) = 50% precisely.

[ QUOTE ]
A coin is bent in such a way that guarantees flips will always be the same (always heads, or always tails). The bent coin is then flipped. What is the probability that the bent coin will be heads?

[/ QUOTE ]

You could then take this abstract model and test it with physical coins, recording statistics and making the same kinds of calculations as I described above for testing the physical "fair" coin. I would expect such a test to produce the same kinds of results and you would end up settling on the assumption that physical operation of the model has not significantly changed it and you have enough confidence it is producing close enough to a Fair result that you could assume the result is fair for your purposes. In this case you would end up assuming P(Heads on First Flip) = 50%.

Now we come to the Real Life situation where jason has told us he has a bent coin sitting on his desk. He asks us to answer the question,

What is P(Heads on First Flip)?

First of all, which type of probability is he asking us for? Is he asking us for the unknown to us long run intrinsic probability that we can estimate statistically with repeated flips of the coin? Or is he asking us to look at the First Flip similiar to the outcomes of First Flips you described according to your Randomizing Bending Process? This is the Crux of the Philisophical debate jason has talked about. Which of these has the most right to be called
P(Heads on First Flip)?

If you think it's the Former then your answer will be, "Not enough information". If you think it's the Latter then you have to make certain assumptions to answer 50%. I think Sklansky has indicated he doesn't care about the philosophy. All he cares about is gambling on the proposition and he solves that easily by just refusing to answer the question and randomizing his call of the coin. I don't think jason has even given an opinion except to say there is great controversy among experts in the field as to which approach should be prefered and that picking one over the other is part of a philisophical debate rather than a mathematical one.

I've basically been providing observations about some of the practical difficulties applying the Later (Baysian) approach in more general settings. I'm giving the warning that if you take that approach be prepared to do so very carefully because your prior conditions may not always be so clear and assumptions should not be casually jumped to about them. They are often One Time events for which imaginary preconditions can become intractably fantastic at times.

For this One Time event of a Bent Coin Jason has sitting on his desk, I suspect he knows that Sklansky has a penchant for picking Heads when calling coin flips and jason has bent the coin to come up Tails. Of course, David has foiled his plans by choosing to randomize his pick instead.

I think most of the disagreements going on here are just due to people not understanding what's being said.

PairTheBoard

jason1990

06-06-2007, 08:29 AM

[ QUOTE ]
This obviously requires further reading on my part.

[/ QUOTE ]
May I suggest that you begin by rereading some of these threads? You have misread some important things. For instance, I said that you subscribed to the philosophy of logical (or objective epistemic) probability, as described in the Wikipedia article. I did not say that de Finetti did. You should reread the quote I gave. Or better yet, click the link and the read the article. Or better yet, go through f97tosc's posting history and find the article he referenced by Jaynes. Look up the article and read that. Or read up on Jaynes himself, maybe starting with http://en.wikipedia.org/wiki/E.T._Jaynes .

There are connections between the claims you are making and the philosophy of de Finetti. But I suspect it will require more studying than you are prepared for in order to understand it.

However, no matter how much you study, you will never be able to experimentally prove someone wrong who says their best guess is 0.4. You can define the word "probability" so that they are wrong according to your definition. You can argue that your definition is the most sensible and most useful. (For an example of how to do this intelligently, go reread your H-bomb.) But you cannot prove you are right with a scientific experiment. If you think you can, then good for you. A lot of folks out here in cyberspace think they can trisect the angle, or create a perpetual motion machine. So you will have sympathizers. If David is unable to find someone he trusts to talk to about this, such as his friend Persi, then he will be right there beside you. You, he, and all the provers of the Riemann hypothesis can post argumentative nonsense all day long.

But if you are truly interested in this subject, then you should temporarily abandon any attempt to connect your ideas with a scientific experiment. Focus, for now, on understanding the philosophy itself by reading the literature. Once you have a foundation of knowledge, return to this question of experiment and falsifiability, and hopefully you will see it more clearly.

jason1990

06-06-2007, 08:30 AM

From David Sklansky (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10678680&an=0&page=0 &vc=1), who likes to start new threads:

[ QUOTE ]
If the only information you have is the number of choices, your personal probability should be equally divided among them. Perhaps because you have data that shows that when the only information is the number of choices the choices have come up equally. But in this particular case even if you don't have the data it is silly to say that you are falling back on subjective judgement. Rather you are falling back on not being a moron.

[/ QUOTE ]
From http://www.philosophyprofessor.com/philosophies/indifference-principle.php :

[ QUOTE ]
principle of indifference

(16th century)

The fundamental principle of statistical theory that unless there is a reason for believing otherwise, each possible event should be regarded as equally likely.

In this crude form, the principle leads to paradoxes because we can group the alternatives in different ways: the next flower I meet might be blue or red, so its being blue has a probability of one-half; but it also might be blue or crimson or scarlet, so the probability of blue is only one-third).

Evidently we require not mere absence of knowledge of reasons favoring one alternative over another, but knowledge of the absence of such reasons. But this may be hard to achieve, even in apparently symmetrical cases like the outcomes of throwing a die; for example, what do we do about the possibility of its standing on edge, or the fact that the paint on the 'six' side will be heavier than on the 'one' side?

[/ QUOTE ]
From http://mathworld.wolfram.com/PrincipleofInsufficientReason.html :

[ QUOTE ]
Principle of Insufficient Reason

A principle that was first enunciated by Jakob Bernoulli which states that if we are ignorant of the ways an event can occur (and therefore have no reason to believe that one way will occur preferentially compared to another), the event will occur equally likely in any way.

Keynes (1921, pp. 52-53) referred to the principle as the principle of indifference, formulating it as "if there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability." Keynes strenuously opposed the principle and devoted an entire chapter of his book in an attempt to refute it.

[/ QUOTE ]
Clearly, Keynes was a moron. From http://plato.stanford.edu/entries/epistemology-bayesian/ :

[ QUOTE ]
Objective Bayesians are the intellectual heirs of the advocates of a Principle of Indifference for probability. Rosenkrantz builds his account on the maximum entropy rule proposed by E.T. Jaynes. The difficulties in formulating an acceptable Principle of Indifference have led most Bayesians to abandon Objective Bayesianism.

[/ QUOTE ]
Clearly, most Bayesians are morons. From http://plato.stanford.edu/entries/probability-interpret/ :

[ QUOTE ]
Bose-Einstein statistics, Fermi-Dirac statistics, and Maxwell-Boltzmann statistics each arise by considering the ways in which particles can be assigned to states, and then applying the principle of indifference to different subdivisions of the set of alternatives, Bertrand-style. The trouble is that Bose-Einstein statistics apply to some particles (e.g. photons) and not to others, Fermi-Dirac statistics apply to different particles (e.g. electrons), and Maxwell-Boltzmann statistics do not apply to any known particles. None of this can be determined a priori ... Critics accuse the principle of indifference of extracting information from ignorance. Proponents reply that it rather codifies the way in which such ignorance should be epistemically managed -- for anything other than an equal assignment of probabilities would represent the possession of some knowledge. Critics counter-reply that in a state of complete ignorance, it is better to assign vague probabilities (perhaps vague over the entire [0, 1] interval), or to eschew the assignment of probabilities altogether.

[/ QUOTE ]
Clearly, all of these philosophers, physicists, and mathematicians are simply morons who do not have the mental clarity of David Sklansky, whose perspective on this is so obviously indisputable, it does not even merit a discussion. If only they would learn a little more MSL. Then they could rise to the top 3% and join David Sklansky in the big leagues.

Phil153

06-06-2007, 08:40 AM

Jason,

Learn to state your point clearly in future. You have failed miserably in this regard, and your bent coin example (as opposed to my dice example) only served to further confuse things.

[ QUOTE ]
However, no matter how much you study, you will never be able to experimentally prove someone wrong who says their best guess is 0.4. You can define the word "probability" so that they are wrong according to your definition.

[/ QUOTE ]
You try to frame this debate as a profound philosophical agreement - in reality it is a dumb semantic debate caused by someone who chose not to (or could not) be precise in his language. For an example of precision in language, read f97tosc's post.

If you are truly interested in this subject, then you should temporarily abandon any attempt to argue this position. Focus, for now, on understanding the philosophy of communication itself by reading the literature. Once you have a foundation of knowledge, return to this thread, and reread your points.

Cheers.

djames

06-06-2007, 09:13 AM

[ QUOTE ]
Jason,
Learn to state your point clearly in future. You have failed miserably in this regard, and your bent coin example (as opposed to my dice example) only served to further confuse things.

[/ QUOTE ]
I'm virtually certain that these lunatic declarations are now read by all as ignorant ramblings. What once may have been read as entertaining nonsense from a devil's advocate, has now certainly lost its humor.

[ QUOTE ]
You try to frame this debate as a profound philosophical agreement - in reality it is a dumb semantic debate caused by someone who chose not to (or could not) be precise in his language.

[/ QUOTE ]

Correct! His name is David Sklansky. You haven't helped much in this regard either even with your clear as mud dice example.

I must say though, you truly did quite appropriately choose your avatar.

jason1990

06-06-2007, 09:23 AM

[ QUOTE ]
I'm virtually certain that these lunatic declarations are now read by all as ignorant ramblings.

[/ QUOTE ]
I think it took me too long to figure that out. It is now time to put him in the box with TOmCowley. Maybe the two of them can sort this out on their own.

Phil153

06-06-2007, 09:35 AM

djames,

You appear not to understand. Jason himself clearly shows that this is a semantic debate by this comment:

[ QUOTE ]
However, no matter how much you study, you will never be able to experimentally prove someone wrong who says their best guess is 0.4. You can define the word "probability" so that they are wrong according to your definition. You can argue that your definition is the most sensible and most useful.

[/ QUOTE ]
How is it possible to have a meaningful discussion when he hasn't even defined the terms he's using? He agrees there's a problem with the definition, but instead of clearly defining the boundaries of that problem, he claims that all "probability" (note that he never defines the term) where we don't have all the information is completely subjective.

Which is an amazing claim, and also false.

Good luck with your disdain for those that disagree with you.

PairTheBoard

06-06-2007, 11:11 AM

[ QUOTE ]

[ QUOTE ]
However, no matter how much you study, you will never be able to experimentally prove someone wrong who says their best guess is 0.4. You can define the word "probability" so that they are wrong according to your definition. You can argue that your definition is the most sensible and most useful.

[/ QUOTE ]
How is it possible to have a meaningful discussion when he hasn't even defined the terms he's using? He agrees there's a problem with the definition, but instead of clearly defining the boundaries of that problem, he claims that all "probability" (note that he never defines the term) where we don't have all the information is completely subjective.

[/ QUOTE ]

The boundries of what problem? Defining "probability"? Have you done that? That's exactly what he has been doing, discussing those boundries as they are understood by experts in the field. Do you want to restrict the boundries to your definition in this particular case? Is that the "problem" you are referring to? What is your defintion? Why should the boundry be restricted to that? When are you going to define Your terms?

[ QUOTE ]

Good luck with your disdain for those that disagree with you.

[/ QUOTE ]

I don't think your comments actually rise to the level of a disagreement. They are more like the wails of a baby. Disagreeable but incoherent as to what your disageement is.

PairTheBoard

Phil153

06-06-2007, 12:05 PM

PTB,

Here's how I see the debate. Please correct me if there's something I am not understanding.

Firstly, my definition of probability: An estimate of the likely distribution of outcomes from some event.

There is no such thing as "actual" probability. Any statement of probability necessarily holds some things constant, or else it would require understanding of the entire universe. For example, in the coin toss example, it is assumed that the action of tossing a coin has no effect on its outcome - i.e. everything is purely random up to the point the coin leaves the fingers. It is assumed that God isn't flipping the coin over when it's about to land on tails. And so on.

Within these restrictions, we can give an estimate of the frequency of a particular outcome occuring. Thus, both Jason and I would say that a symmetrical coin made of homogeneous material has a 50% chance of landing on heads. We would both agree that this is an objective estimate based on the known information (symmetrical coin, homogeneous material, meaning no bias based on the laws of physics) and assumptions (God not messing with us). If you do not agree with this, please say so.

The point is that we do not have all the relevant information, ever. Every estimate is made on the partial information we have.

When Jason introduces an unknown bias into the coin (removing our knowledge of symmetry), Jason claims we can make no estimate of the frequency of outcomes from tossing this coin. Or that such estimates are purely subjective.

And I disagree with him. We can make claims that are objectively more or less reasonable in the long run. For example, to claim that the coin will be 50% likely to land on heads or tails is a superior statement to "the coin is 95% likely to land on heads". He claims these statements are objectively equivalent. And I disagree. Because we do have the information that there should be no particular reason for the coin to be bias specifically toward heads or tails. Just like we have the information that God isn't watching out coin tosses and flipping over the tails. You can see this testing out more than one situation with a bent coin. My dice example makes this point a lot clearer.

Jason seems incapable of making his point clear to intelligent lay people, which is usually a sign of fuddled thinking. So if you, PTB, can tell me what I'm missing in Jason's statements, then it would be appreciated.

I do understand your points about bias in choosing information and bias in assuming, and I agree that these are dangers. But these difficulties do not mean the process is either subjective or not falsifiable.

chezlaw

06-06-2007, 12:18 PM

[ QUOTE ]
What is P(Heads on First Flip)?

[/ QUOTE ]

hate to throw a further coin in the fountain but at the deep level this thread has plunged don't we have the problem that they're all first flips as the coin is not identically the same coin after its flipped as it was before.

So either we can extrapolate from information about other coins to objectively predict future behavior or we can't. This is true of all flips of all coins.

but its hard to see why this matters in practice. Is DS's method used correctly ever going to create a situation where you'd be correct to bet against him?

chez

KipBond

06-06-2007, 12:32 PM

[ QUOTE ]
Thus, both Jason and I would say that a symmetrical coin made of homogeneous material has a 50% chance of landing on heads.
...
When Jason introduces an unknown bias into the coin (removing our knowledge of symmetry), Jason claims we can make no estimate of the frequency of outcomes from tossing this coin. Or that such estimates are purely subjective.

And I disagree with him. We can make claims that are objectively more or less reasonable in the long run. For example, to claim that the [bent] coin will be 50% likely to land on heads or tails is a superior statement to "the coin is 95% likely to land on heads".

[/ QUOTE ]

Would you agree that there must be some difference in the two bolded probabilities? If we say a (almost perfectly) fair coin has a 50% probability of landing on heads, it sure seems much more true than if we say a bent coin that biases one side more than the other has a 50% probability of landing on heads.

In the 2nd case, we know we are ignorant of some crucial information; specifically, the exact bias that has been introduced into the coin. In the 1st, there is no such reasonable knowledge of ignorance.

So... that's why I introduced my "accuracy" of probabilities, which PTB then clarified/expounded as being "confidences". So, we are reasonably confident in our fair coin probability assessment, and we should be reasonably unconfident(*) in our bent coin probability assessment.

(*) Apparently "unconfident" is not a word?!? (http://dictionary.reference.com/browse/unconfident) Who knew?!?

KipBond

06-06-2007, 12:35 PM

[ QUOTE ]
don't we have the problem that they're all first flips as the coin is not identically the same coin after its flipped as it was before.

[/ QUOTE ]

PTB mentioned that (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10676292):

[ QUOTE ]
About all you can do at the end of all this is assume the coin has not been significantly altered by the flips and that you have been convinced from the statistical evidence that it's close enough to Fair to assume it is Fair for your purposes, like maybe gambling with it.

[/ QUOTE ]

PairTheBoard

06-06-2007, 01:03 PM

[ QUOTE ]
Firstly, my definition of probability: An estimate of the likely distribution of outcomes from some event.

[/ QUOTE ]

What is the "distribution of outcomes" for a one time event? For the bent coin, Which "outcomes", plural, are you talking about? We are only looking at the First Flip. Do you intend to make additional Flips of the coin and look at those as "outcomes"? Define your "outcomes" in the Bent Coin case whereby there is some "distribution" for them which you judge to be "likely".

[ QUOTE ]
we can give an estimate of the frequency of a particular outcome occuring.

[/ QUOTE ]

"Frequency" with respect to what repeated experiment? Tossing the same Bent Coin repeatedly? Or flipping this particular Coin For the First Time over and over???

[ QUOTE ]
When Jason introduces an unknown bias into the coin (removing our knowledge of symmetry), Jason claims we can make no estimate of the frequency of outcomes from tossing this coin. Or that such estimates are purely subjective.

And I disagree with him. We can make claims that are objectively more or less reasonable in the long run. For example, to claim that the coin will be 50% likely to land on heads or tails is a superior statement to "the coin is 95% likely to land on heads".

[/ QUOTE ]

Why? What do you mean by "likely"? What "long run" are you talking about? This is a one time event. We are only going to Flip this particular bent coin for the First Time Once. So where is your "long run". "Long run" with respect to what? Frequency with respect to what "outcomes"? Flipping the coin repeteadyly? Or flipping it for the First Time over and over???

If you mean flipping the Coin repeatedly to estimate it's propensity for coming up heads, your 50% estimate is just a guess. Why is it the "superior" guess? Because it's in the middle of all other guesses? Your 50% guess is likely to not even be a close estimate for what the statistics will show with repeated flips of this particular bent coin.

Do you want to say your 50% is the "best" estimate? According to what mathematics do you measure "best"? How do you intend to apply this guess in such a way that your judgment of what's the "best" guess gives you an advantage over other guesses? Is that how you intend to restrict the discussion of "best"? What about other ways we might need to apply our guess whereby guessing 50% is not actually "best". Do you intend that 50% is "best" for all applications of our guess? Did you read the examples by djames? That's what he was talking about if you didn't understand him.

PairTheBoard

chezlaw

06-06-2007, 01:05 PM

[ QUOTE ]
[ QUOTE ]
don't we have the problem that they're all first flips as the coin is not identically the same coin after its flipped as it was before.

[/ QUOTE ]

PTB mentioned that (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10676292):

[ QUOTE ]
About all you can do at the end of all this is assume the coin has not been significantly altered by the flips and that you have been convinced from the statistical evidence that it's close enough to Fair to assume it is Fair for your purposes, like maybe gambling with it.

[/ QUOTE ]

[/ QUOTE ]
Txs, don't know how I missed that /images/graemlins/wink.gif

but isn't that a huge problem? to get probabilities about further flips from earlier flips requires exactly the Baysian techniques that are being derided.

chez

Phil153

06-06-2007, 01:13 PM

PTB,

"Best" means "most likely to have the smallest deviation from the actual* value". See my dice example. This is also the reason why 50% heads is superior to 99% heads, regardless of how much this particular coin is skewed.

As for the long run, you know exactly what that is. Chezlaw nailed it.

[ QUOTE ]
Why? What do you mean by "likely"? What "long run" are you talking about? This is a one time event.

[/ QUOTE ]
Everything is a one time event, but some types of events get repeated, and show trends. It is on these trends, and only on these trends, that we base all probability estimates, including the flip of a regular minted coin or one that's been bent an unknown amount in an unknown direction.

This fact is the only thing that allows us to say anything at all about the universe, or the probability of an actual outcome. Again, see Chezlaw.

* "actual" means "the point at which we know everything except that which we hold constant or unfailingly random".

Phil153

06-06-2007, 01:19 PM

Kip,

I agree with the confidence/unconfidence thing, obviously. I've used the term myself - it goes directly to my point.

I think the whole problem with this thread is that Jason hasn't defined what he's arguing. I suspect when he does it will evaporate like the Aether.

wtfsvi

06-06-2007, 01:21 PM

ptb. If I flip a fair coin it will land either heads or tails based on what side I start it on, what force I apply to it and where I apply this force, to some degree where I am (windy, thin air, so on) and how long I let the coin stay in the air. None of these conditions are impossible to know, yet you don't know them or how they affect the outcome. Still I get the impression you agree that there is a 50% chance of heads if I flip a fair coin right now, probably based on the assumption that there is a random distribution to all the factors I mentioned above. Why do you selectively apply the indiference principle?

PairTheBoard

06-06-2007, 01:29 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
don't we have the problem that they're all first flips as the coin is not identically the same coin after its flipped as it was before.

[/ QUOTE ]

PTB mentioned that (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10676292):

[ QUOTE ]
About all you can do at the end of all this is assume the coin has not been significantly altered by the flips and that you have been convinced from the statistical evidence that it's close enough to Fair to assume it is Fair for your purposes, like maybe gambling with it.

[/ QUOTE ]

[/ QUOTE ]
Txs, don't know how I missed that /images/graemlins/wink.gif

but isn't that a huge problem? to get probabilities about further flips from earlier flips requires exactly the Baysian techniques that are being derided.

chez

[/ QUOTE ]

Nobody has derided Baysian techniques here chez. We have repeatedly confirmed that they can be useful. What I have said is that they can also be tricky and misleading if you are not careful with them. Do you consider that to be "deriding"? Did you notice that in my description of the statistical tools that can be applied to the data of repeated flips I gave equal treatment to both Baysian and Conventional techniques?

I'm not sure the final assumption that the coin does not get physically altered by the flips has anything to do with Baysian techniques. It's merely a practical assumption that allows us to put the computations we have made from an Ideal Mathematical Model into practical use. How do you see that as being "Baysian"?

PairTheBoard

PairTheBoard

06-06-2007, 01:44 PM

[ QUOTE ]
"Best" means "most likely to have the smallest deviation from the actual* value".

[/ QUOTE ]

How do you measure that? What is your mathematical Model for measureing that? What measure do you use for "deviation"? Simple Difference? Or the commonly used L2 norm, ie. square of the difference. Are you measuring the Average Difference? Average with respect to what presumed probability distribution for Bent Coin biases?

[ QUOTE ]
As for the long run, you know exactly what that is. Chezlaw nailed it.

[/ QUOTE ]

I don't know that Chezlaw "nailed" anything. I'm talking to you now.

[ QUOTE ]
Everything is a one time event, but some types of events get repeated, and show trends. It is on these trends, and only on these trends, that we base all probability estimates, including the flip of a regular minted coin or one that's been bent an unknown amount in an unknown direction.

[/ QUOTE ]

So what Events are getting repeated to show a "trend" for the One Time event of flipping this particular Bent Coin for the First Time?

PairTheBoard

djames

06-06-2007, 01:49 PM

A few golden items from Phil:

[ QUOTE ]
"Best" means "most likely to have the smallest deviation from the actual* value".

[/ QUOTE ]

[ QUOTE ]
* "actual" means "the point at which we know everything except that which we hold constant or unfailingly random".

[/ QUOTE ]

Say what?

[ QUOTE ]
Jason seems incapable of making his point clear to intelligent lay people, which is usually a sign of fuddled thinking.

[/ QUOTE ]

If you replace Jason with Phil, I think you have a bingo. Your posts seem even more cryptic & misleading than David's.

After these goodies, there's really no point in me explaining to you that "best" should imply that an optimization has occurred. I can't even figure out what metric you've chosen let alone if it will properly reach your objective once optimized.

PairTheBoard

06-06-2007, 01:57 PM

[ QUOTE ]
Kip,

I agree with the confidence/unconfidence thing, obviously. I've used the term myself - it goes directly to my point.

I think the whole problem with this thread is that Jason hasn't defined what he's arguing. I suspect when he does it will evaporate like the Aether.

[/ QUOTE ]

I don't think jason is even making an argument. He is describing the state of affairs here which you, and Sklansky want to ignore evidently because it is over your heads. You ignore them at your own peril just like the Baseball manager who ignores Statistics so he can make decisions by the seat of his pants. If you would ever do the work of actually being precise in defining your own terms you might be able to understand what the discussion is about.

PairTheBoard

TomCowley

06-06-2007, 01:57 PM

[ QUOTE ]
Bose-Einstein statistics, Fermi-Dirac statistics, and Maxwell-Boltzmann statistics each arise by considering the ways in which particles can be assigned to states, and then applying the principle of indifference to different subdivisions of the set of alternatives, Bertrand-style. The trouble is that Bose-Einstein statistics apply to some particles (e.g. photons) and not to others, Fermi-Dirac statistics apply to different particles (e.g. electrons), and Maxwell-Boltzmann statistics do not apply to any known particles. None of this can be determined a priori

[/ QUOTE ]

If you're going to namedrop einstein, or quote somebody to namedrop einstein, at least make sure you have a point. Both F-D and B-E statistics are based completely on the indifference principle. In BOTH, the likelihood of occupation of all states of equal energy is the same. The difference is simply what configurations are legal, not the application of the indifference principle.

The only "point" your quote makes is that you have to have a number of outcomes (about which you know nothing else probabilistic) to assign a probability of 1/N to each of them. Derf. It's not a profound insight nor anything Sklansky would ever argue with.

Phil153

06-06-2007, 02:04 PM

[ QUOTE ]
I don't think jason is even making an argument.

[/ QUOTE ]
And therein lies the problem. He is the voice of doubt which never quantifies the bounds of those doubts. Just like you. Where he has claimed something (such as probability estimates being subjective, implying that 0.5 is not better than 0.99 for a bent coin estimate), it has been false.

Phil153

06-06-2007, 02:05 PM

[ QUOTE ]
The only "point" your quote makes is that you have to have a number of outcomes (about which you know nothing else probabilistic) to assign a probability of 1/N to each of them. Derf. It's not a profound insight nor anything Sklansky would ever argue with.

[/ QUOTE ]
Actually, it's precisely what Sklansky argues for. And Jason is disagreeing with, in his bent coin example and claims of subjectivity.

PairTheBoard

06-06-2007, 02:16 PM

[ QUOTE ]
ptb. If I flip a fair coin it will land either heads or tails based on what side I start it on, what force I apply to it and where I apply this force, to some degree where I am (windy, thin air, so on) and how long I let the coin stay in the air. None of these conditions are impossible to know, yet you don't know them or how they affect the outcome. Still I get the impression you agree that there is a 50% chance of heads if I flip a fair coin right now, probably based on the assumption that there is a random distribution to all the factors I mentioned above. Why do you selectively apply the indiference principle?

[/ QUOTE ]

At some point I have to make an assumption so that I can put probability to work. I freely admit I'm making the assumptions you are describing. One reason I find these assumptions reasonable is because they are falsifable by experimental testing and statistical analysis on those tests. In this case I can repeatedly flip the coin. But I hesitate to make such assumptions where there is no way to test them empirically.

For example, What is the probability for Life on Mars? The assumptions I would need to make about imaginary universes become so fanciful and beyond any means of empirical testing that I hesitate to apply that approach. The Bent Coin example is right on the cusp of these kinds of questions. Your universe of imaginary coins being bent at random seems close to one you might model and test according to your model. But whatever Model we use, we don't know if it accurately describes the process by which the particular coin on Jason's desk got bent.

So it's a good example to look at. It forces us to think hard about what we are doing here. We should be able to see how the same principles apply even though our intuition tries to lead us away from looking at them.

PairTheBoard

Phil153

06-06-2007, 02:22 PM

[ QUOTE ]
[ QUOTE ]
"Best" means "most likely to have the smallest deviation from the actual* value".

[/ QUOTE ]

How do you measure that? What is your mathematical Model for measureing that? What measure do you use for "deviation"? Simple Difference? Or the commonly used L2 norm, ie. square of the difference. Are you measuring the Average Difference?

[/ QUOTE ]
Any is fine for the bent coin example /images/graemlins/smile.gif

[ QUOTE ]
Average with respect to what presumed probability distribution for Bent Coin biases?

[/ QUOTE ]
One based on indifference between choosing a bend toward heads or tails. Just like your regular coin toss is based on indifference by God, the minting process, and the laws of physics.

[ QUOTE ]
So what Events are getting repeated to show a "trend" for the One Time event of flipping this particular Bent Coin for the First Time?

[/ QUOTE ]
We can randomly bend 100 coins. This is equivalent to the situation where coins are only bent one way but we have zero information about it. It's easy to get confused on this point when you're zipping in and out of reference frames without realizing it. Just like in relativity, a probability estimate can be objectively correct in two reference frames but not in a third.

What am I saying? That for any given set of information, we can produce a "most objective" estimate of probability, and correctly apply the indifference principle to that which we do not know.

PairTheBoard

06-06-2007, 02:33 PM

[ QUOTE ]
[ QUOTE ]
I don't think jason is even making an argument.

[/ QUOTE ]
And therein lies the problem. He is the voice of doubt which never quantifies the bounds of those doubts. Just like you. Where he has claimed something (such as probability estimates being subjective, implying that 0.5 is not better than 0.99 for a bent coin estimate), it has been false.

[/ QUOTE ]

First of all, you quoted me out of context. Seocnd of all, you still haven't defined your terms.

PairTheBoard

PairTheBoard

06-06-2007, 02:36 PM

[ QUOTE ]
[ QUOTE ]
The only "point" your quote makes is that you have to have a number of outcomes (about which you know nothing else probabilistic) to assign a probability of 1/N to each of them. Derf. It's not a profound insight nor anything Sklansky would ever argue with.

[/ QUOTE ]
Actually, it's precisely what Sklansky argues for. And Jason is disagreeing with, in his bent coin example and claims of subjectivity.

[/ QUOTE ]

This is out of context with the "outcomes" I was talking about in my discussion with you.

PairTheBoard

djames

06-06-2007, 02:40 PM

[ QUOTE ]
[ QUOTE ]
Bose-Einstein statistics, Fermi-Dirac statistics, and Maxwell-Boltzmann statistics each arise by considering the ways in which particles can be assigned to states, and then applying the principle of indifference to different subdivisions of the set of alternatives, Bertrand-style. The trouble is that Bose-Einstein statistics apply to some particles (e.g. photons) and not to others, Fermi-Dirac statistics apply to different particles (e.g. electrons), and Maxwell-Boltzmann statistics do not apply to any known particles. None of this can be determined a priori

[/ QUOTE ]

If you're going to namedrop einstein, or quote somebody to namedrop einstein, at least make sure you have a point. Both F-D and B-E statistics are based completely on the indifference principle. In BOTH, the likelihood of occupation of all states of equal energy is the same. The difference is simply what configurations are legal, not the application of the indifference principle.

The only "point" your quote makes is that you have to have a number of outcomes (about which you know nothing else probabilistic) to assign a probability of 1/N to each of them. Derf. It's not a profound insight nor anything Sklansky would ever argue with.

[/ QUOTE ]

Why did you reply to me, but quote someone else?

Phil153

06-06-2007, 02:46 PM

Precisely what terms haven't I defined?

PairTheBoard

06-06-2007, 03:19 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
"Best" means "most likely to have the smallest deviation from the actual* value".

[/ QUOTE ]

How do you measure that? What is your mathematical Model for measureing that? What measure do you use for "deviation"? Simple Difference? Or the commonly used L2 norm, ie. square of the difference. Are you measuring the Average Difference?

[/ QUOTE ]
Any is fine for the bent coin example /images/graemlins/smile.gif

[ QUOTE ]
Average with respect to what presumed probability distribution for Bent Coin biases?

[/ QUOTE ]
One based on indifference between choosing a bend toward heads or tails. Just like your regular coin toss is based on indifference by God, the minting process, and the laws of physics.

[/ QUOTE ]

Ok. So now we have some mathematics we can work with. The prior distibution for the parameter p=P(Heads) that Baysians would normally use based on the indifference principle is the Uniform Distribution on [0,1]. It is symmetric as you desire. I assume you prefer the standard metric for measuring error of |.5-p| where p is a random variable with that Uniform Distribution. And you do claim that you want to Average that measured error of |.5-p| over that Distribution for p.

Evidently you did not read and understand djames' posts above where he pointed out that when you measure that Average you get exactly the same result as you would for any other guess. The average of |1.0-p| over that Distribution for p is exactly the same as the average of |.5-p| or |.4-p| or |.1-p|. You didn't know that did you phil? That's because you have never done the work to define your terms well enough to know what you are talking about. Then you claim jason is not being clear?

[ QUOTE ]
[ QUOTE ]
So what Events are getting repeated to show a "trend" for the One Time event of flipping this particular Bent Coin for the First Time?

[/ QUOTE ]
We can randomly bend 100 coins. This is equivalent to the situation where coins are only bent one way but we have zero information about it. It's easy to get confused on this point when you're zipping in and out of reference frames without realizing it. Just like in relativity, a probability estimate can be objectively correct in two reference frames but not in a third.

[/ QUOTE ]

You have identified your own confusion because that is exactly what you have been doing, "zipping in and out of reference frames without realizing it".

You say, "We can randomly bend 100 coins". Ok. You are at least working toward something that can be mathematically modeled. Except, how do you define "random". Your definition of "random" determines the prior distribution for the parameter p=P(Heads) that I discussed above. By "random" do you mean for your test to produce a Uniform Distribution for p? If so, your test only confirms my observation above that your guess of .5 is no better than 1.0, .4, or .1 for its Average absolute difference to the outcomes of your test. If you had read djames and clairified your thinking you would have already known this. How else have you failed to clarify your thinking in trying to understand what jason has said? You don't even know do you?

[ QUOTE ]

What am I saying? That for any given set of information, we can produce a "most objective" estimate of probability, and correctly apply the indifference principle to that which we do not know.

[/ QUOTE ]

The trouble is you haven't thought about this in enough depth, or read the thinking of qualified people on this point to where you might realize that different people can evaluate the information differently. You hypothesize the existence of an Ideal "Perfect" evalution. Good luck proving that.

PairTheBoard

PairTheBoard

06-06-2007, 03:30 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Bose-Einstein statistics, Fermi-Dirac statistics, and Maxwell-Boltzmann statistics each arise by considering the ways in which particles can be assigned to states, and then applying the principle of indifference to different subdivisions of the set of alternatives, Bertrand-style. The trouble is that Bose-Einstein statistics apply to some particles (e.g. photons) and not to others, Fermi-Dirac statistics apply to different particles (e.g. electrons), and Maxwell-Boltzmann statistics do not apply to any known particles. None of this can be determined a priori

[/ QUOTE ]

If you're going to namedrop einstein, or quote somebody to namedrop einstein, at least make sure you have a point. Both F-D and B-E statistics are based completely on the indifference principle. In BOTH, the likelihood of occupation of all states of equal energy is the same. The difference is simply what configurations are legal, not the application of the indifference principle.

The only "point" your quote makes is that you have to have a number of outcomes (about which you know nothing else probabilistic) to assign a probability of 1/N to each of them. Derf. It's not a profound insight nor anything Sklansky would ever argue with.

[/ QUOTE ]

Why did you reply to me, but quote someone else?

[/ QUOTE ]

Because the person he is quoting has him on Ignore for the same reasons I do. He told us to F-off. We did.

PairTheBoard

jason1990

06-06-2007, 04:31 PM

[ QUOTE ]
The prior distibution for the parameter p=P(Heads) that Baysians would normally use based on the indifference principle is the Uniform Distribution on [0,1].

[/ QUOTE ]
f97tosc has pointed out here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10322109&page=0&vc=1 ) that Jaynes recommends against the uniform prior in this situation. If we really know nothing, he recommends the (improper) prior whose density is 1/(x(1-x)) on [0,1]. The Jeffreys prior is yet another non-uniform prior recommended in this "no information" setting. Even among those who would seek out a "most objective" estimate of probability for a given set of information, there is no consensus over what prior to use here. Readers should understand that this is a highly nontrivial topic. Real debates on this topic do not involve simple proclamations of "this is obvious" or "My way works. Period." And I do not think the scientists involved are calling each other morons.

Incidentally, there is a nice discussion of these various priors here (http://en.wikipedia.org/wiki/Prior_probability#Uninformative_priors).

P.S. Some readers may wonder, what do these "priors" have to do with anything? Well, that is Part 2 of this debate. I told you about the bent penny on my desk, and there has been all this back and forth about the probability of heads. But suppose I told you I just flipped it and it came up heads. What is the probability of heads on the second flip? You no longer have "no information." What is your "most objective" estimate given this one piece of information? Is it unique? Is there room for anyone to disagree with you? Do you think you can prove your guess is the "best" with some kind of scientific experiment? Do you think you can prove it with a philosophical argument? Do you even know how to apply Bayes Theorem in this situation? Can you identify the assumptions you are making when you apply it? All of this relates to these priors and has already been brought up by f97tosc earlier in this thread.

TomCowley

06-06-2007, 05:46 PM

None of your proclamations are ever more useful than nihilism. (and as for who I respond to, I hit the nearest reply button- it's the same message at the bottom of the same thread). You've accepted that 0.5 is the proper number to use for a money line. Show ANY practical example, with no bias-creating information beyond "two possible outcomes", where using a different estimate of P(heads) is demonstrably better for ANY purpose, excluding cases where you would not use an estimate of P(heads)=0.5 if you knew with 100% certainty that P(heads) was uniform on [0,1]

PLOlover

06-06-2007, 06:28 PM

Not that I'm educated or anything, but doesn't information theory say that when you make your guess you add information into the system?

What I mean is that if you randomize your guess then basically you randomize the whole system, so the actual coin doesn't even matter, whether it's fair or always heads or always tails.

I mean since the coin and your guess is symmetric (heads,tails) , it seems that only one needs to be random, either your guess or the coin, for the whole thing to be random or to be break even.

I mean you can look at it as a two way deal, does the coin result match the guess, or does the guess match the coin result.

If the coin is fair you can break even by always guessing tails or always guessing heads or any combo, because the coin flip randomizes the system, but if the coin flip is not random then you must randomize your guess to have a break even system.

Phil153

06-06-2007, 06:52 PM

PTB,

It's obvious that if you pick an even distribution the [0,1] line then any guess is as good as any other when taking the simple average of all cases. But I am not claiming an even distribution on the [0,1] line. I am claiming indifference in the bend direction. The slight bend may be peaking around the extremes of [45,55], in which case 99% heads is a pretty awful guess. That is why the 50% figure is superior in many situations that will come up (but not all - in others it won't matter). Note that the binary nature of the coin toss hides this fact from being intuitively/logically obvious, which is why my dice example is clearer.

Here is how Jason originally posed the bent coin problem:

[ QUOTE ]
Imagine that I take a quarter and I bend it with pliers. I do not show you the quarter. I force you to choose a number for the probability that this coin will come up heads. You can do no better than to choose 50%. But you are almost certainly wrong. By saying 50%, you are saying the coin is fair. The coin is almost certainly not fair. A smarter answer would be to say, "there is not enough information."

[/ QUOTE ]

And here's you:

[ QUOTE ]
Then you claim jason is not being clear?

[/ QUOTE ]

This is what I'm talking about with Jason's flitting in and out of reference frames. The last bolded statement is flat out false. He confuses the "actual" probability with a best estimated probability and uses the difference to show our best estimated model is flawed or inferior to "not enough information". Which it isn't. His failure to state his case clearly (a case he may well have, if it was stated properly) has led to much of the confusion in this thread.

When someone as smart/educated as yourself in math* strongly disagrees with me, I do a Sklansky and push up the odds I am wrong. Hence, I really do have to read some literature as well as these threads very closely before responding.

PLOlover

06-06-2007, 07:36 PM

[ QUOTE ]

Here is how Jason originally posed the bent coin problem:

Quote:
Imagine that I take a quarter and I bend it with pliers. I do not show you the quarter. I force you to choose a number for the probability that this coin will come up heads. You can do no better than to choose 50%. But you are almost certainly wrong. By saying 50%, you are saying the coin is fair. The coin is almost certainly not fair. A smarter answer would be to say, "there is not enough information."

[/ QUOTE ]

I think Jason doesn't realize that saying "not enough info" with respect to the coin and 50% with respect to ;your guess are totally consistent with each other.
as a matter of fact if know the coin is 50% then your guess is "any" or "doesn't matter" or "irrelevant info" in regards to breaking even.

see my post above.

chezlaw

06-06-2007, 08:36 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
don't we have the problem that they're all first flips as the coin is not identically the same coin after its flipped as it was before.

[/ QUOTE ]

PTB mentioned that (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10676292):

[ QUOTE ]
About all you can do at the end of all this is assume the coin has not been significantly altered by the flips and that you have been convinced from the statistical evidence that it's close enough to Fair to assume it is Fair for your purposes, like maybe gambling with it.

[/ QUOTE ]

[/ QUOTE ]
Txs, don't know how I missed that /images/graemlins/wink.gif

but isn't that a huge problem? to get probabilities about further flips from earlier flips requires exactly the Baysian techniques that are being derided.

chez

[/ QUOTE ]

Nobody has derided Baysian techniques here chez. We have repeatedly confirmed that they can be useful. What I have said is that they can also be tricky and misleading if you are not careful with them. Do you consider that to be "deriding"? Did you notice that in my description of the statistical tools that can be applied to the data of repeated flips I gave equal treatment to both Baysian and Conventional techniques?

I'm not sure the final assumption that the coin does not get physically altered by the flips has anything to do with Baysian techniques. It's merely a practical assumption that allows us to put the computations we have made from an Ideal Mathematical Model into practical use. How do you see that as being "Baysian"?

PairTheBoard

[/ QUOTE ]
Its only by applying baysian reasoning that we can reasonably conclude that a flip tells us anything about further flips. So either we use basysian methods, or all flips are first flips and the method of repeated flip tells us nothing about the probable results of the next flip.

I take back the word 'deride' it wasn't correct. I'm trying to understand how there can be any substantive criticism of DS's method when there's no disagreement with the results and no means of calculating probabilities in the real world without it.

I can kinda see your concerns about it causing confusion if used carelessly but it's also a very simple way of looking at things and pushes us to consider the a very important factor in applications which is what information is available to us and others.

chez

taobot

06-06-2007, 08:41 PM

[ QUOTE ]
[ QUOTE ]

Here is how Jason originally posed the bent coin problem:

Quote:
Imagine that I take a quarter and I bend it with pliers. I do not show you the quarter. I force you to choose a number for the probability that this coin will come up heads. You can do no better than to choose 50%. But you are almost certainly wrong. By saying 50%, you are saying the coin is fair. The coin is almost certainly not fair. A smarter answer would be to say, "there is not enough information."

[/ QUOTE ]

I think Jason doesn't realize that saying "not enough info" with respect to the coin and 50% with respect to ;your guess are totally consistent with each other.
as a matter of fact if know the coin is 50% then your guess is "any" or "doesn't matter" or "irrelevant info" in regards to breaking even.

see my post above.

[/ QUOTE ]

Now the bent coin gets thrown 100 times. You will win a prize if you correctly guess the number of times that the coin will come up heads. What would you answer?

PLOlover

06-06-2007, 09:15 PM

[ QUOTE ]
Now the bent coin gets thrown 100 times. You will win a prize if you correctly guess the number of times that the coin will come up heads. What would you answer?

[/ QUOTE ]

hmm, well if you're looking for a break even then still a random guess would do it assuming you get true odds, 99-1.

If you're getting something free then any guess you make is +EV.

game theory only allows you to make choices that can't be exploited, it doesn't guarantee you the best choice or the biggest edge. (obviously a fair coin you should guess around 50 and a bent coin you should bet 50+-x if you knew the bent direction and how much)

PairTheBoard

06-06-2007, 09:54 PM

[ QUOTE ]
It's obvious that if you pick an even distribution the [0,1] line then any guess is as good as any other when taking the simple average of all cases. But I am not claiming an even distribution on the [0,1] line. I am claiming indifference in the bend direction. The slight bend may be peaking around the extremes of [45,55], in which case 99% heads is a pretty awful guess.

[/ QUOTE ]

Now you're doing more than just defining your terms. You are making an assumption. This is why it's so important to state your assumptions along with the Conclusion that depends on them. We want to look at your assumptions to see if they are reasonable. We want to compare them to other assumptions. Notice that before, all you asked for in the distruibution was that it be...

[ QUOTE ]
One based on indifference between choosing a bend toward heads or tails.

[/ QUOTE ]

The Uniform, or "Even" as you put it, distribtion satisfies that criteria. It wasn't so obvious to you before that it didn't support your Conclusion. Not only does the Uniform distribution satisfy your criteria but it is the Standard one used by Baysians in applying the criteria of "indifference" to these kinds of problems.

But you don't Know your assumption is correct do you? Why should you think it's more correct than assuming the Uniform? Do you think Jason is not strong enough to bend the sh-t out of that coin with a pair of pliers? Do you think he is not inclined to do so? His inclination may be to completely fold it over with essentially only one side left to it. If he did this with indifference to Heads or Tails then guessing 1.0 or 0 would be just as good guesses as 0.5 by the average of absolute difference measure you have agreed to.

So we can argue about which assumption for the prior distribution of Bent Coin biases is most realistic. The standard Baysian is Uniform. As jason's recent post points out there is heavy non trivial disagreement with that. You appear to stand in the camp of adopting more of an 1/(x)(1-x) model which you think brings in your common knowledge about how coins are likely to be bent. You could even set up a machine to bend them according to that distribution.

But we are not dealing with how coins are likely to be bent by arbitrary people under imaginary conditions. We are dealing with the One and Only Bent Coin of jason's. And we have no idea what he had in mind when he bent it, do we?

[ QUOTE ]
[ QUOTE ]
Here is how Jason originally posed the bent coin problem:

[/ QUOTE ]

Imagine that I take a quarter and I bend it with pliers. I do not show you the quarter. I force you to choose a number for the probability that this coin will come up heads. You can do no better than to choose 50%. But you are almost certainly wrong. By saying 50%, you are saying the coin is fair. The coin is almost certainly not fair. A smarter answer would be to say, "there is not enough information."

[/ QUOTE ]

[ QUOTE ]
[ QUOTE ]
And here's you:

Then you claim jason is not being clear?

[/ QUOTE ]

This is what I'm talking about with Jason's flitting in and out of reference frames. The last bolded statement is flat out false.

[/ QUOTE ]

Is he the one flitting with reference frames or are you? jason's statment was perfectly clear to me. Why wasn't it to you? Because you want to talk about some other reference frame than what he is talking about? You want That reference frame to define P(heads)? Does your reference frame limit itself to this specific coin sitting on jason's desk? Or does your reference frame talk about a bunch of imaginary coins with an imaginary bending process producing a distribution for Bent Coin biases which is not Uniform as the Standard Baysian assumption would be but symmetric with a hump in the middle? Why did you think jason was Ever talking about That reference frame? Because that's how you intuit what the probability statement should mean?

He has a coin on his desk which he personally has Bent with a pair of pliers. What is the probability That Coin comes up Heads when flipped? Not the probabilty that a random coin produced by your random model comes up heads on the first flip. Just the probability this specific coin "comes up heads". The reference frame he is talking about is clear, especially in the context of his other comments. You are the one who wantd to flit to the other reference frame and claim it provides the true definition for this probability. Fine, go ahead and flit. We can talk about that reference frame if you want to. But you are the one introducing it as the proper definition for this probability. A major point in this discussion is the fact that not everybody thinks that is the proper reference frame for defining this probability.

And even if we do go to your reference frame, it is only in your idealized model that P(Heads on First Flip)=50% is guaranteed to be correct. In the reality of a world of different people bending coins with a pair of pliers I imagine they do produce an asymmetry with respect to Heads and Tails. I think people would have a bias bending the coin with Heads facing them, pressing one side down against the table and folding the coin up towards the heads side, thus making it easier to land Heads when flipped. So even in the referernce frame you introduce, in reality your P(Heads on First Flip) = 50% is still almost certainly wrong. People almost certainly have some psychological bias in reality.

[ QUOTE ]
He confuses the "actual" probability with a best estimated probability and uses the difference to show our best estimated model is flawed or inferior to "not enough information". Which it isn't. His failure to state his case clearly (a case he may well have, if it was stated properly) has led to much of the confusion in this thread.

[/ QUOTE ]

For the reference frame he is clearly talking about, wouldn't you agree that the "A smarter answer would be to say, there is not enough information"? You are the one who has confused his reference frame for the one you want to introduce to define this "probability". And you are the one bringing in your common knowledge as an assumption for your reference frame.

He said, "You can do no better than to choose 50%" if you are forced to give a number in his reference frame for what defines this probability. Would you disagree with that? That is exactly the logic asserted in f97tosc's H-bomb post for a situation of "total lack of information other than symmetry", which you seemed to understand so well when you labeled it an H-bomb.

Did you understand the rest of the H-bomb post when he talked about what happens when you no longer have total ignorance of information or when you apply "common knowledge" to the information you do have? Did you understand his explanation of possible prior distributions for the Bent Coin's bias. You should have because that is what you are applying when you talk about 50% being the "best estimate".

We just went through that didn't we? It turns out your "best estimate" depends on assumptions about a prior distribution which you have brought in according to what you think is common knowledge about the situation. f97tosc understood what jason was saying by, ""You can do no better than to choose 50%" and f97tosc was agreeing with jason. You would have realized that if you had understood either one of them.

You say, "His failure to state his case clearly (a case he may well have, if it was stated properly) has led to much of the confusion in this thread."

Who is the one who has failed to state his case clearly? Who is the one who had to be forced to define his terms? Who is the one who had to have his Assumptions based on his purported common knowledge dragged out of him?

phil, jason is a highly trained professional who makes his living based on clear thinking and mathematical research in the field of probability which must pass extensive scrutiny in peer review for clarity and soundness of logic. When questioning who is failing to provide clarity of thought here I suggest you take a look at yourself first.

PairTheBoard

jason1990

06-06-2007, 10:15 PM

[ QUOTE ]
I take back the word 'deride' it wasn't correct. I'm trying to understand how there can be any substantive criticism of DS's method when there's no disagreement with the results and no means of calculating probabilities in the real world without it.

[/ QUOTE ]
I got a call the other day from a woman named Marion, who works for the local Indian tribe. They are building a casino and want someone from our department to advise them as they inspect their roulette wheels and train their dealers. They want to be certain that the numbers on the wheel are really equally likely to come up. She had it in mind that we would do some series of statistical tests or something. But maybe I should tell her that I know a shortcut. I will tell her that David Sklansky says they will be just fine as long as they have no information. When she takes me to see the roulette tables, I will shout at her, "Don't look at it. Shut your eyes, Marion, and don't look at it, no matter what happens. If you look at it, you might get some information and throw the probabilities out of whack."

The point of this little fable, of course, is that there is a big practical difference between equal likelihood that someone might claim on the basis of ignorance, and equal likelihood that has been established by statistical testing. The statistical testing methods can be classical or Bayesian, as PairTheBoard has discussed. In the Bayesian case, if you do enough tests, it will not really matter what your prior was. With each test, you will update the prior with real data. After enough updates, your subjective prior will be "drowned out" by all the real data.

Someone mentioned "confidence". David says the first flip of the bent coin has probability 50% of coming up heads. If the coin were not bent, he would say the same thing. But with an unbent coin, he would be more confident in this claim. He has no confidence in his claim for the bent coin. That is why he must use his fair coin to choose heads or tails for him. He uses the fair coin to avoid taking the worst of it, because he recognizes that he really does not know the "true" probability of heads.

If he were to flip the coin 100,000 times, then each time he could update his opinion with real data. At the end, he could be very confident in his final probability. And in the end, it would hardly matter at all whether he used the indifference principle or not to start his analysis.

But if he flips it only once, that is a whole different story. If he updates his prior only once, and tries to estimate the probability of heads on the second flip, he will be hardly any more confident of this estimate than he was of his original estimate. With only one flip, it is his prior that will drown out the data.

Now think back to where this all began (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10508631&an=&page=2& vc=1). We had a probability model involving a defendant and his shoe size. We imposed a prior probability distribution on his shoe size which was based on ignorance. We then got one piece of data. And we updated our prior. Once. Like flipping the coin once. The posterior is a mix of the prior (which we chose out of ignorance) and this one piece of data. Which weighs more? Does the subjective prior drown out the one data point, like we would expect it to do in the case of the bent coin? This is not necessarily an easy question to even formulate, let alone answer. But it is definitely something that should be considered, especially in light of the fact that the indifference principle is surrounded by philosophical controversy. If I were the defendant and David was on the jury, I would want this controversy brought to the attention of the other jurors. I would want them to realize that this indifference principle David appeals to is not the slam dunk triviality he makes it out to be. And I would not want him to push them around and call them morons just because they might choose to disagree with something that a lot of great minds in this century have disagreed with.

Phil153

06-06-2007, 10:17 PM

PTB,

My "indifferent to heads or tails" model (which is the only thing I've ever claimed) is a far larger subset and requires far less in the way of assumptions than the uniform [0,1] range which you ridiculously apply to this situation.

DO YOU SEE WHY?

[ QUOTE ]
The standard Baysian is Uniform.

[/ QUOTE ]
The standard Baysian distribution has nothing to do with this situation or any of my claims

DO YOU SEE WHY?

[ QUOTE ]
Who is the one who has failed to state his case clearly? Who is the one who had to be forced to define his terms? Who is the one who had to have his Assumptions based on his purported common knowledge dragged out of him?

[/ QUOTE ]
It's very obvious to me that Jason's original scenario is very poorly constructed where it isn't flat out wrong. I CHOSE to define my terms to get out this hopeless semantic mess that Jason started, in the hopes that Jason would follow...apparently not.

[ QUOTE ]
phil, jason is a highly trained professional who makes his living based on clear thinking and mathematical research in the field of probability which must pass extensive scrutiny in peer review for clarity and soundness of logic. When questioning who is failing to provide clarity of thought here I suggest you take a look at yourself first.

[/ QUOTE ]
Do you know him personally? Is it possible that this is clouding your judgment here? I asked a friend of mine (math grad) to look at this thread and he agrees with my position 100%.

If Jason would just come out and frame his position clearly (and perhaps give a more compelling or enlightening example), I doubt we would have a debate.

I'm done with this nonsense until either Jason makes a post where he clearly states his case, or my private readings give me some deeper insight into his and your positions in this thread.

PLOlover

06-06-2007, 10:26 PM

[ QUOTE ]
David says the first flip of the bent coin has probability 50% of coming up heads.

[/ QUOTE ]

not true at all.

jason1990

06-06-2007, 10:33 PM

[ QUOTE ]
[ QUOTE ]
David says the first flip of the bent coin has probability 50% of coming up heads.

[/ QUOTE ]

not true at all.

[/ QUOTE ]
Oh, right, I forgot. He avoided that question. Here is the revised paragraph:

Someone mentioned "confidence". David says if the only information you have is the number of choices, your personal probability should be equally divided among them. David's personal probability for the chance of heads in the bent coin is therefore 50%. If the coin were not bent, he would say the same thing. But with an unbent coin, he would be more confident in this claim. He has no confidence in his claim for the bent coin. That is why he must use his fair coin to choose heads or tails for him. He uses the fair coin to avoid taking the worst of it, because he recognizes that he really does not know the "true" probability of heads.

Phil153

06-06-2007, 10:41 PM

I posted just after you did. Bringing this back into concrete terms (the jury/shoe size example) would make things clearer for us lay folk. And help exorcise the demons of fuzzy thinking.

PairTheBoard

06-06-2007, 10:54 PM

[ QUOTE ]
Its only by applying baysian reasoning that we can reasonably conclude that a flip tells us anything about further flips. So either we use basysian methods, or all flips are first flips and the method of repeated flip tells us nothing about the probable results of the next flip.

[/ QUOTE ]

I think you are confusing the application of Bayes' Theorem in a mathematical probability model to the Philosophy of Baysian Probabilists. Nobody has any criticisms or even caveats for appying Bayes' Theorem in such a model.

However, that is not what we are referring to when we talk about Baysian techniques in Statistics. Not long ago, jason started an excellent thread in the Probability Forum where he made a great case for applying such techniques. How best to estimate your poker Win Rate playing at a new location based on limited data taking into account your known Win Rate at a previous location where the games were more favorable?

He compared the results obtained from Baysian techniques to those of standard statistics which would treat the new data without regard to previous historical data. He made a very good case for why we should prefer the Baysian techniques. And the result was practical for people making bankroll/limits decisions when switching from Party Poker to Full Tilt for example.

[ QUOTE ]
I take back the word 'deride' it wasn't correct. I'm trying to understand how there can be any substantive criticism of DS's method when there's no disagreement with the results and no means of calculating probabilities in the real world without it.

[/ QUOTE ]

There is no criticism of David's method of flipping a fair coin to make the Call for flip of the Bent Coin. But the general principle he keeps harping on says that given two Choices with no information about them, A and B, they are equally likely to be true. The principle is useless because he is talking about an A,B which are indistinguishable.

We don't know what they are. He could have said B,A just as easily and we couldn't tell the difference. Once you actually identify A and B you now do have information about them. Do you want to judge their comparative liklihood based on that information or do you want to bias your judgement based on his principle for when there was "no information"? Why don't you read the link jason provided in his recent post which explains the logical difficulties in that?

Sklansky continues to promote this on his latest Thread as something only a moron would ignore. Does he think he's the only one who has ever thought of it? Will he read up on what experts think of it? No. By refusing to clarify his notion he can continue to maintain that he knows he's right whether he is making any sense or not. Read the comment by piers as well as mine on that thread.

[ QUOTE ]
I can kinda see your concerns about it causing confusion if used carelessly but it's also a very simple way of looking at things and pushes us to consider the a very important factor in applications which is what information is available to us and others.

[/ QUOTE ]

I agree that Baysian techniques in Statistics can be very useful. See my description above of jason's thread in the Probability Forum that makes exactly that case in a very practical situation for poker players. You should be able to find it with a little effort on the Search function. He convince me.

PairTheBoard

djames

06-06-2007, 11:12 PM

[ QUOTE ]
I asked a friend of mine (math grad) to look at this thread and he agrees with my position 100%.

[/ QUOTE ]

Best of luck to your friend. May I suggest a non-mathematical career?

TomCowley

06-06-2007, 11:16 PM

The only actual "issue" that I'm aware of in the jury case is that DS assumed P(size13|suspect) = P(size13) in the general population, and there is no direct reason why this should be true. If there were another partial footprint that was clearly from a size 12-14 shoe, all suspects would have that shoe size range, and the proper probability to use would be P(size 13|size 12-14) which isn't nearly as incriminating.

Instead of stating this correction to Sklansky's model in a coherent form, like "You need to consider the correlation of the footprint to prior evidence", which is obviously true, and something DS would surely agree with, PTB went off on his philosophical nihilist nonsense about how you can never know the degree of correlation exactly, so this type of analysis is not valid. And suggesting fuzzy human thinking is superior to Bayesian analysis using best estimates of correlation and probability.

PairTheBoard

06-06-2007, 11:16 PM

[ QUOTE ]
My "indifferent to heads or tails" model (which is the only thing I've ever claimed) is a far larger subset and requires far less in the way of assumptions than the uniform [0,1] range which you ridiculously apply to this situation.

[/ QUOTE ]

That's debatable. You know nothing about measuring "size of set" of this kind. It remains an Assumption of yours. DUCY?

[ QUOTE ]
The standard Baysian distribution has nothing to do with this situation or any of my claims

[/ QUOTE ]

That's debatable and still your Assumption. DUCY?

[ QUOTE ]
It's very obvious to me that Jason's original scenario is very poorly constructed where it isn't flat out wrong. I CHOSE to define my terms to get out this hopeless semantic mess that Jason started, in the hopes that Jason would follow...apparently not.

[/ QUOTE ]

His original scenario is well constructed and his statements are correct. You are the one who has supplied the semantical difficulties and additional assumptions. DUCY?

[ QUOTE ]
Do you know him personally? Is it possible that this is clouding your judgment here? I asked a friend of mine (math grad) to look at this thread and he agrees with my position 100%.

[/ QUOTE ]

I don't know jason personally and only know him from his posts here and on the Probability Forum. Of course there's always a chance my judgement is cloudy for any number of possible reasons. But on this matter I take my judgement over yours and over this unknown friend of yours who I know practially nothing about. If your friend decides to post here I will certainly listen to what he has to say. And we're still waiting for Persi Diaconis to show up.

[ QUOTE ]
I'm done with this nonsense until either Jason makes a post where he clearly states his case, or my private readings give me some deeper insight into his and your positions in this thread.

[/ QUOTE ]

After you've done your research you may realize that it is you who has been providing the nonsense. Good reading.

PairTheBoard

PLOlover

06-07-2007, 12:21 AM

[ QUOTE ]
Quote:

Quote:
David says the first flip of the bent coin has probability 50% of coming up heads.

not true at all.

Oh, right, I forgot. He avoided that question. Here is the revised paragraph:

Someone mentioned "confidence". David says if the only information you have is the number of choices, your personal probability should be equally divided among them. David's personal probability for the chance of heads in the bent coin is therefore 50%. If the coin were not bent, he would say the same thing. But with an unbent coin, he would be more confident in this claim. He has no confidence in his claim for the bent coin. That is why he must use his fair coin to choose heads or tails for him. He uses the fair coin to avoid taking the worst of it, because he recognizes that he really does not know the "true" probability of heads.

[/ QUOTE ]

you guys arent looking at this right. theres the flip (heads,tails) and the guess (heads,tails).

DS is saying that the guess should be heads 50% of the time, he's not saying anything about the flip.

PairTheBoard

06-07-2007, 03:18 AM

[ QUOTE ]
[ QUOTE ]
Quote:

Quote:
David says the first flip of the bent coin has probability 50% of coming up heads.

not true at all.

Oh, right, I forgot. He avoided that question. Here is the revised paragraph:

Someone mentioned "confidence". David says if the only information you have is the number of choices, your personal probability should be equally divided among them. David's personal probability for the chance of heads in the bent coin is therefore 50%. If the coin were not bent, he would say the same thing. But with an unbent coin, he would be more confident in this claim. He has no confidence in his claim for the bent coin. That is why he must use his fair coin to choose heads or tails for him. He uses the fair coin to avoid taking the worst of it, because he recognizes that he really does not know the "true" probability of heads.

[/ QUOTE ]

you guys arent looking at this right. theres the flip (heads,tails) and the guess (heads,tails).

DS is saying that the guess should be heads 50% of the time, he's not saying anything about the flip.

[/ QUOTE ]

David's latest thread where he makes yet another attempt to clarify his postition is almost a case study in vagueries and ill defined terms. What does he mean by "personal probability" as it appears in bold above and as he presented it in that thread? His statement came in the context of these,

[ QUOTE ]
DS -
Probability is related to the information you have about a subject. Two people can have two different assessments of a situation because they have different information. There is no "right" answer. Is the next card turned going to be the ace of spades. You will give a probability estimate based on what cards you have seen. Even a fair coin is not even money when flipped if you have some physical information about the flipper.

When the information is not as oviously correlated with probability as seen cards, you resort to your knowledge of that information. A black and a white horse are about to race. You know nothing else about them. If you happen to have knowledge of other races involving black and white horses, say the black horse won 45,000 out of 100,000 you should use that ratio.

[/ QUOTE ]

So he is talking about "probability" as it relates to information. He points out that your probability may differ from mine because we have different information. Thus your probability is personal to the information you have. It's clear then that he's talking about your evalution of the probability based on your information. From his Cards example he indicates that if you know there are 7 flush cards left out of 42 unseen cards, your probability for the flush card coming will be 1/6 while another player's probability may be different based on what cards he has seen. Sklansky would say that 1/6 is your personal probability for the Flush card coming. That's the probability you would use if you want to bet on the flush. He certainly doesn't mean that you would roll a die and choose to bet on the flush when a six comes on the die.

The same with his Black-White horse example. You would bet on the black horse according to the information you have that gives you a 45% estimate for its probability for winning. You certainly wouldn't spin a 100 slot wheel with 45 out of 100 black slots and decide to bet on the black horse when your wheel tells you to.

So in that context I think it's clear what he means by, if the only information you have is the number of choices, your personal probability should be equally divided among them.. In the Bent Coin case he means to say that you should give both Heads and Tails your personal probability estimate based on "no information" of 50% for each. Nothing there about randomizing how you call the flip. He means to say that you can choose to call heads using your personal probability estimate P(heads)=50% based on your personal condition of "no information but the two choices Heads or Tails". This at least is what anyone reading his latest clarification thread would have to conclude is what he means by "personal probability".

To be clear, when I say P(heads)=50% I mean the probability of heads for the bent coin on the first flip based on no information but the two choices heads, tails. How will David use this probability? He will use it the same way he uses 1/6 for the flush and 45% for the Black horse when making gambling decisions. But he can't gamble with himself. He must gamble with someone else who has the same information as he does. So we must assume David's gambling opponent has the same information David does. That rules out David's gambling with Jason because Jason bent the coin and knows how he bent it. It rules out gambling with anybody who might have personal insight into Jason.

But he can gamble with somebody who has the exact same information as he does and if that person, call him Mickey, offers him 1000-999 odds David would have to declare that to be a favorable bet for himself.

And he would be right if the information known to both parties was only that there are two choices A and B. This was pointed out by f97tosc. But is that really the only information available? We have the additional information that a human being who both David and Mickey know equally well has bent the coin with a pair of pliers. The information does not really satisfy f97tosc's criteria that it give "no hint" to preference for Heads over Tails. The hint may be hard to see. But phil has claimed that this information has provided him with an insight into likely distributions for the parameter p=Long Run Probability of Heads for this Coin. phil claims that distribution is obviously not Uniform.

What insights does Mickey see in that "hint" of information? Mickey may decide he sees the likelihood that human beings tend to bend coins with pliers using the Heads "front" side of the coin face up when they bend it. If he is correct, David has overestimated the lack of information here and may very well take the worst of it when he accepts Mickey's 1000-999 proposition.

This illustrates a problem with David's principle. It only applies perfectly in the pure Theoretical case of two Unknown choices A,B. There's no way we can test it in the Theoretical case because there's no way for us to distingish between two unknown choices. How could he be proven wrong?

What he really means to say is that if we are specified two choices and we can see no hint for choosing one over the other then we should estimate them to be "close to" equally likely. He can really only say "close to" because he has no way of knowing if we are "seeing" all there is to see in the information provided by specifying the choices. As a general principle to teach people this might have some wings. But stating it clearly also points out its pitfalls. Are we "seeing" everything?

Futhermore, how else might Sklansky use his "no information" estimate for P(heads)=50%? We see him on other threads applying this principle for drawing further conclusions that are not strictly gambling on the one time event. He applies it as a Truth probability for statements. He applies it as a prior estimate bias when evaluting probabilities after actual information has been obtained. In other words, he biases known information with the No-Information that came before it. He dodges this with the Black-White horses example by swamping his No-Information bias with plenty of new information.

But watch out for what he does when the new information is not so plentiful. Like jason pointed out, how would he apply it after just One Flip of the Biased Coin? That is a deep question hotly debated. What does Sklansky say about that? In his latest clarification he indicates that subjective methods would be used. He doesn't detail what subjective method he would use after One Flip of the Bent Coin. So it's hard to tell if he is using logically consistent methods in general. Especially when he applies this No-Information bias to Sparse Known information in some of his other arguments on other topics. Some of which have only the most obscure relationship to gambling on cards, horses, or Bent Coins.

PairTheBoard

PLOlover

06-07-2007, 03:50 AM

[ QUOTE ]
He must gamble with someone else who has the same information as he does. So we must assume David's gambling opponent has the same information David does.

[/ QUOTE ]

not at all. as long as you are getting true odds it doesn't matter, that's the point.
I mean, suppose you write down either 1 or 0. Now you show it to a 3rd party, no hanky panky or anthing going on. It's a fair game. you know which number yo wrote down, but I don't. As long as the bets are even money there can be no advantage as long as I choose 1 as often as I choose 0.

PairTheBoard

06-07-2007, 03:57 AM

[ QUOTE ]
[ QUOTE ]
He must gamble with someone else who has the same information as he does. So we must assume David's gambling opponent has the same information David does.

[/ QUOTE ]

not at all. as long as you are getting true odds it doesn't matter, that's the point.
I mean, suppose you write down either 1 or 0. Now you show it to a 3rd party, no hanky panky or anthing going on. It's a fair game. you know which number yo wrote down, but I don't. As long as the bets are even money there can be no advantage as long as I choose 1 as often as I choose 0.

[/ QUOTE ]

I suggest you read my post again and think about it for a while.

PairTheBoard

mosta

06-07-2007, 12:15 PM

[ QUOTE ]
It should be pretty clear that for a single flip the bent and non-bent coin pay-odds are identical. However, consider the following scenarios:

case A:
You flip a fair coin 1,000,000 times in a row, getting 11:10

case B:
You flip a bent coin, of unknown bias 1,000,000 times in a row, getting 11:10.

You do not know your results until after completing all of your flips.

With the fair coin it doesn't matter what you do, you can predict the likelihood of various results using a normal distribution. With the bent coin the distribution is anormal, and the strategy you employ will effect the distribution. Right?

[/ QUOTE ]

I'm not going to read the whole thread. But this idea above occurred to me at first. but then I decided it wasn't right.

it's still the same with multiple flips. yes your results will be of much greater magnitude, and you may end up getting "raped" by guessing wrong. but you could just as well have hit the jackpot by guessing right. multiplying the magnitude of the game doesn't change the point. if you are told the game will be for 100 flips, and you have to stay with your initial choice, it's going to hurt a lot if you picked the bent side. but the other possible occasions when you may guess the unbent side will offset that with a correspondingly big score. think of the repetition as other occasions to play with different bent coins. (note that you need not have an actual chance to ever play again. as long as you could have guessed the side that always wins just as well as the side that always loses, 50% is fair value. (ie, as long as he doesn't cheat by adjusting the coin against your choice.))

and this is probably addressed in the big thread below, but I think it's safe to assert that if you do get to change your choice after any throw, you should always pick the prior (or most frequent prior) result. that's kind of a "duh." but is probably a good rule of thumb for any bet with any coin without knowing if it's fair.

chezlaw

06-07-2007, 01:10 PM

[ QUOTE ]
Quote:
--------------------------------------------------------------------------------

Its only by applying baysian reasoning that we can reasonably conclude that a flip tells us anything about further flips. So either we use basysian methods, or all flips are first flips and the method of repeated flip tells us nothing about the probable results of the next flip.

--------------------------------------------------------------------------------

I think you are confusing the application of Bayes' Theorem in a mathematical probability model to the Philosophy of Baysian Probabilists. Nobody has any criticisms or even caveats for appying Bayes' Theorem in such a model.

[/ QUOTE ]
Much danger of me getting confused. I'm just stuck in a groove where it seems DS's method has no actual problems in practice and is a neccesary way of making use of probabilities in practice. I don't understand why he want to deny events have actual probabilities but it seems not to matter. I'm vey willing to accept it might matter but I haven't understood why it does (except your point about it being confusing).

DS is talking is about decisions not probabilities. When making a decision all that matters is what information you have - including information about what information other people have and what information might be obtainable.

Then its not that actual probabilities of events don't exist but: when making a decision it doesn't matter what the actual probability of the event is but what information you have. No information and it doesn't matter which decision you make.

chez

PairTheBoard

06-07-2007, 02:40 PM

[ QUOTE ]
DS is talking is about decisions not probabilities. When making a decision all that matters is what information you have - including information about what information other people have and what information might be obtainable.

Then its not that actual probabilities of events don't exist but: when making a decision it doesn't matter what the actual probability of the event is but what information you have. No information and it doesn't matter which decision you make.

[/ QUOTE ]

That sounds nice. But he tries to encode that idea into the mathematical framework of probability models. Once he starts doing that we have every right to ask if he's doing it in a way that makes sense according to the way mathematical probability works. If he is doing it in especially bad ways he will end up using the math to make conclusions that may very well be bogus. When couched in the aura of the mathematical language we end up seeing bogus conclusions carrying the bogus air of mathematical authority.

PairTheBoard

chezlaw

06-07-2007, 02:59 PM

[ QUOTE ]
[ QUOTE ]
DS is talking is about decisions not probabilities. When making a decision all that matters is what information you have - including information about what information other people have and what information might be obtainable.

Then its not that actual probabilities of events don't exist but: when making a decision it doesn't matter what the actual probability of the event is but what information you have. No information and it doesn't matter which decision you make.

[/ QUOTE ]

That sounds nice. But he tries to encode that idea into the mathematical framework of probability models. Once he starts doing that we have every right to ask if he's doing it in a way that makes sense according to the way mathematical probability works. If he is doing it in especially bad ways he will end up using the math to make conclusions that may very well be bogus. When couched in the aura of the mathematical language we end up seeing bogus conclusions carrying the bogus air of mathematical authority.

PairTheBoard

[/ QUOTE ]
I'm not sure this isn't all a confusion of language rather than some substantive disagreement. Maybe we can get a clarification /images/graemlins/grin.gif

chez

jason1990

06-07-2007, 02:59 PM

[ QUOTE ]
When making a decision all that matters is what information you have

[/ QUOTE ]
The information does not always translate into a unique, mathematically indisputable decision. For the bent coin, I will give you one piece of information: the first flip was heads. Now here is your decision problem: should you accept 11 to 10 odds that the second flip will be tails? Assume no trickery; for instance, you are wagering with someone who has the same information as you.

The decision you make will have to be based not only on your one piece of information, but also on your prior function f(P). The math is a little technical, but suffice it to say that the prior function is not uniquely determined by the information or lack thereof, even by people who believe the same things David does.

chezlaw

06-07-2007, 03:17 PM

[ QUOTE ]
[ QUOTE ]
When making a decision all that matters is what information you have

[/ QUOTE ]
The information does not always translate into a unique, mathematically indisputable decision. For the bent coin, I will give you one piece of information: the first flip was heads. Now here is your decision problem: should you accept 11 to 10 odds that the second flip will be tails? Assume no trickery; for instance, you are wagering with someone who has the same information as you.

The decision you make will have to be based not only on your one piece of information, but also on your prior function f(P). The math is a little technical, but suffice it to say that the prior function is not uniquely determined by the information or lack thereof, even by people who believe the same things David does.

[/ QUOTE ]
I'm going to grope a litle bit here as I'm not quite sure what you mean, but it seems a nice simple example to work with.

Is this prior function some summation of the information we have about coins, how bending effect them and how this particular coin came to be in its present state?

Does it include information about the other person willing to take the other side of the bet or is that accounted for elsewhere?

chez

jason1990

06-07-2007, 03:28 PM

This is from a post I made (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10698435&page=0&vc=1 ) in luckyme's new thread:

[ QUOTE ]
f97tosc talked about "priors" here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10653713&page=0&vc=1 ):

[ QUOTE ]
Suppose that we write f(P) for the prior probability distribution that the coin is such that when tossed, it will result in heads a long-term fraction P of the time.

[/ QUOTE ]
The function f(P) is the prior. Essentially, P is the probability of heads for the bent coin, and f(P) is your estimate of the probability that the coin is actually bent that way. For example, if you think all bends are equally likely, then the function f(P) is flat. If you think extreme bends are less likely than small bends, then the function f(P) has a hump in the middle.

[/ QUOTE ]
If you want further clarification, then ask away. But this can be a start.

chezlaw

06-07-2007, 03:48 PM

[ QUOTE ]
This is from a post I made (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10698435&page=0&vc=1 ) in luckyme's new thread:

[ QUOTE ]
f97tosc talked about "priors" here (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10653713&page=0&vc=1 ):

[ QUOTE ]
Suppose that we write f(P) for the prior probability distribution that the coin is such that when tossed, it will result in heads a long-term fraction P of the time.

[/ QUOTE ]
The function f(P) is the prior. Essentially, P is the probability of heads for the bent coin, and f(P) is your estimate of the probability that the coin is actually bent that way. For example, if you think all bends are equally likely, then the function f(P) is flat. If you think extreme bends are less likely than small bends, then the function f(P) has a hump in the middle.

[/ QUOTE ]
If you want further clarification, then ask away. But this can be a start.

[/ QUOTE ]
So isn't that prior function a result of information about coins and how they come to be bent etc?

chez

jason1990

06-07-2007, 04:00 PM

[ QUOTE ]
So isn't that prior function a result of information about coins and how they come to be bent etc?

[/ QUOTE ]
There is a group of mathematicians who want to codify the method of converting information into a prior. The first question they must answer is, how do we convert "no information" into a prior. That is, if you have no information about coins and how they come to be bent, then what prior should you use? First answer this, and then you can incorporate whatever information you might have about coins and how people bend them.

As I was describing in this post (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10685709&page=0&vc=1 ), there is no consensus for how to convert "no information" into a prior in this case. Some may start with a flat prior and then adjust it for whatever they might know about coins and how they are bent. Some may start with a non-flat prior and adjust that. Even though they have the same information about coins and how they are bent, they will end up with different priors, because they started in different places.

PairTheBoard

06-07-2007, 04:02 PM

I guess I should clean up this blunder I made in my discussion with phil.

[ QUOTE ]
PTB -
Ok. So now we have some mathematics we can work with. The prior distibution for the parameter p=P(Heads) that Baysians would normally use based on the indifference principle is the Uniform Distribution on [0,1]. It is symmetric as you desire. I assume you prefer the standard metric for measuring error of |.5-p| where p is a random variable with that Uniform Distribution. And you do claim that you want to Average that measured error of |.5-p| over that Distribution for p.

When you measure that Average you get exactly the same result as you would for any other guess. The average of |1.0-p| over that Distribution for p is exactly the same as the average of |.5-p| or |.4-p| or |.1-p|.

[/ QUOTE ]

That seemed "obvious" to me for some reason, without checking the math. I'm not sure now if djames said that as well or not. Anyway, phil bought it,

[ QUOTE ]
phil -
It's obvious that if you pick an even distribution the [0,1] line then any guess is as good as any other when taking the simple average of all cases.

[/ QUOTE ]

After doing some actual math I see that the average absolute difference between the guess 0.5 and such a randomly bent coin's P(Heads) is 1/4. If you guess 1.0 the average absolute difference is 1/2. Just thinking about it a little makes that pretty "obvious".

So I guess as Jason has pointed out, if all you assume about the prior is symmetry then your "best" guess is 0.5 for the true long run P(Heads) of the specific bent coin coming from a symmetric prior distribution.

Sorry about that phil. See why peer review is so important?

But your humped distribution does come into play for what you do after you see One Flip of the bent coin.

PairTheBoard

Phil153

06-07-2007, 05:16 PM

My point of contention was that you were assuming (with no reason) that an even [0,1] distribution was reasonable from my statement of "indifferent to direction". Which is of course nonsense. Hence I responded:

[ QUOTE ]
It's obvious that if you pick an even distribution the [0,1] line then any guess is as good as any other when taking the simple average of all cases.

[/ QUOTE ]

No, I did not bother to check the spread, because the assumptions going into your analysis had nothing to do with anything. I should have.

[ QUOTE ]
[ QUOTE ]
My "indifferent to heads or tails" model (which is the only thing I've ever claimed) is a far larger subset and requires far less in the way of assumptions than the uniform [0,1] range which you ridiculously apply to this situation.

[/ QUOTE ]

That's debatable. You know nothing about measuring "size of set" of this kind. It remains an Assumption of yours. DUCY?

[ QUOTE ]

The standard Baysian distribution has nothing to do with this situation or any of my claims

[/ QUOTE ]

That's debatable and still your Assumption. DUCY?

[/ QUOTE ]
While we're correcting things...how about you correct the above as well?

The set of all distributions that are symmetrical about the mean are indeed larger than the uniform distribution on [0,1] (since the uniform distribution is a subset of it). It's not debatable at all. I guess you can do the math by integrating the set of symmetrical distributions and comparing it with the single [0,1] set. I don't know how to do this - do you? However, it's not needed unless you want to flex your probability muscles, because logic trumps probability PhDs every times.

Also, Jason made a statement that was blatantly false (it was bolded in a post above), and yet you claim he made no errors and was perfectly clear.

You shut your mind off on this debate a long time ago. Your last few posts were an amazingly closed minded vitriol. But I'd like to know if you now agree that the 0.5 pick is better than any other in the absence of other information about the coin and assuming indifference toward heads or tails. Which was the only thing I ever claimed and amazingly was disagreed with by mathematical "experts".

Your points about assumptions going into a model are trivial and obvious (even where their application is not). I do not think we have a point of contention except for Jason's ridiculous false claims which you continue to support.

[ QUOTE ]
Sorry about that phil. See why peer review is so important?

[/ QUOTE ]
There you go being an arrogant [censored] again. I really don't think there is anything esoteric to this debate - if you follow an unexploitable money line for any of the examples discussed, then that's as deep as you ever need to get for anything in the real world.

jason1990

06-07-2007, 05:19 PM

[ QUOTE ]
So isn't that prior function a result of information about coins and how they come to be bent etc?

[/ QUOTE ]
As a follow up to my last post, let me try to give you a clearer explanation.

As I said, P is the probability of heads for the bent coin, and f(P) is your estimate of the probability that the coin is actually bent that way. Someone else may look at the exact same information about coins and how they come to be bent etc, and come up with a different estimate, for no apparent reason whatsoever. It would be nice if we had a way to determine who was "right". In order to determine who is right, we must find a "recipe" for converting the given information into a prior function f(P).

There is a group of mathematicians who advocate something along the lines of what David is saying. They believe such a recipe exists, and that we can find one which is logically better than any other recipe. If we find it, then we can say who is right. If we find it, then the prior function will depend only on the information itself, and not on who is looking at that information.

Unfortunately, even in the simplest of cases, mathematicians within this group do not agree on what the right recipe is. Their disagreements are not simply technical, semantic nonsense. The disagreeing parties each have solid, logical arguments in support of their favorite recipes. And in complicated cases, there may not be any reasonable recipes at all.

Phil153

06-07-2007, 05:28 PM

You are merely contesting the assumptions going into a model. This is stuff of argument (physics, experience), and has nothing to do with the esoteric probability wars going on. At least not in any of the examples that have been posed so far.

From the link you provided, it seems the questions on which there are no clear answer apply to things like "if we have an unbounded set of unknown distribution, how should we model its distribution in order to get a number?"

I really don't see what that has to do with David's jury thread (where this first came up) or any of the gambling examples given since. These are questions for physics, logic and experience, not esoteric math.

Can you come up with an example of where they apply to the real world situations that David is trying to model?

PairTheBoard

06-07-2007, 05:37 PM

I corrected one mistake phil. When are you going to correct yours?

PairTheBoard

chezlaw

06-07-2007, 05:41 PM

[ QUOTE ]
There is a group of mathematicians who advocate something along the lines of what David is saying. They believe such a recipe exists, and that we can find one which is logically better than any other recipe. If we find it, then we can say who is right. If we find it, then the prior function will depend only on the information itself, and not on who is looking at that information.

[/ QUOTE ]
Is that what DS is saying? I'd assume he would allow that there could be cases where there is no good answer at all.

Doesn't seem to harm the claim that when making a decision it doesn't matter what the actual probability of the event is but what information you have. Its just saying that sometimes we have no way of knowing which are good decisions.

chez

Phil153

06-07-2007, 05:53 PM

[ QUOTE ]
I corrected one mistake phil. When are you going to correct yours?

PairTheBoard

[/ QUOTE ]
I can't correct what I'm not aware of.

Phil153

06-07-2007, 06:20 PM

[ QUOTE ]
Can you come up with an example of where they apply to the real world situations that David is trying to model?

[/ QUOTE ]
Jason's already done this - missed the thread:

http://forumserver.twoplustwo.com/showfl...=0#Post10699986 (http://forumserver.twoplustwo.com/showflat.php?Cat=0&Number=10699986&an=0&page=0#Pos t10699986)

This should be a better discussion (and future discussions on how this affects other situations where there is other information) instead of the nonsense in here.

jason1990

06-07-2007, 06:35 PM

[ QUOTE ]
[ QUOTE ]
There is a group of mathematicians who advocate something along the lines of what David is saying. They believe such a recipe exists, and that we can find one which is logically better than any other recipe. If we find it, then we can say who is right. If we find it, then the prior function will depend only on the information itself, and not on who is looking at that information.

[/ QUOTE ]
Is that what DS is saying? I'd assume he would allow that there could be cases where there is no good answer at all.

[/ QUOTE ]
I thought so too. Way back in the beginning, I said that sometimes it is better to say "not enough information" than to try to assign a probability and compute an EV or apply Bayes theorem. Apparently, he did not agree with that. He also said,

[ QUOTE ]
My point is that there is no such thing as THE probability of an event. There is only a probability correlated with the information you have.

[/ QUOTE ]
and

[ QUOTE ]
Probability is related to the information you have about a subject. Two people can have two different assessments of a situation because they have different information.

[/ QUOTE ]
Notice he did not say, "Two people can have two different assessments of a situation even though they have the same information." I am pretty sure he disagrees with that sentence. After all, he does not believe that probability is subjective, so probability cannot depend on the person, only on the information (according to his words, as I understand them).

This is his definition of probability and he has made claims which suggest this is the only logical or reasonable definition of probability.

His position, therefore, seems pretty clear to me: Information determines probability. Since "not enough information" is never a necessary conclusion, information always determines probability. Since the same information cannot produce two different probabilities, information determines a unique probability.

If you know the probability and you know the odds you are getting, you can always compute an EV and make your decision. So he is asserting (at least the existence of) an algorithm which, in theory, takes the information you have and produces a definitive decision. Not only that, this algorithm is logically superior to any other algorithm, since there is only one unique probability, and therefore only one unique decision, for any given set of information. He may not claim to know that algorithm in all its gory details. But he believes in its existence.

This is what that group of mathematicians would like to develop. But some other mathematicians think their efforts are in vain, because they believe such an algorithm does not exist. You could say they are the "algorithm atheists" and David is an "algorithm theist". There is no proof either way, since no such algorithm has been created and agreed upon, even in the simplest of cases.

PairTheBoard

06-07-2007, 06:59 PM

The thing is, David also likes to apply probabilities for the Truth of Propositions. Wouldn't Godel's Theorems preclude the existence of this All Powerful Algorithm for doing that for all possible Propositions?

PairTheBoard

jason1990

06-07-2007, 07:40 PM

[ QUOTE ]
The thing is, David also likes to apply probabilities for the Truth of Propositions. Wouldn't Godel's Theorems preclude the existence of this All Powerful Algorithm for doing that for all possible Propositions?

[/ QUOTE ]
This is a good point. But I doubt it does. (Now we are really getting esoteric!)

David is not the only one who wants to apply probability to propositions. One thing is that a "proposition", by definition, must have a truth value. It is either true or false. If it is not either true or false, then it is not a proposition.

Using Godel's theorem to produce a counterexample could be viewed as analogous to saying that the algorithm does not exist because it could never compute the probability of "Have a nice day."

KipBond

06-07-2007, 07:43 PM

[ QUOTE ]
I'd like to know if you now agree that the 0.5 pick is better than any other in the absence of other information about the coin and assuming indifference toward heads or tails.

[/ QUOTE ]

How can 0.5 be the best pick if it has a 0% confidence level (& a 0% chance of being correct, assuming a "bent coin" can't be bent in such a way as to make it a fair coin)?

KipBond

06-07-2007, 07:57 PM

[ QUOTE ]
[ QUOTE ]
I'd like to know if you now agree that the 0.5 pick is better than any other in the absence of other information about the coin and assuming indifference toward heads or tails.

[/ QUOTE ]

How can 0.5 be the best pick if it has a 0% confidence level (& a 0% chance of being correct, assuming a "bent coin" can't be bent in such a way as to make it a fair coin)?

[/ QUOTE ]

How about if you are allowed 2 picks prior to any flips of the coin. Then, after your 2 picks, the coin is flipped 1 billion times to find the probability that it will come up heads in any 1 flip. Whichever pick of yours is closest to this probability is then compared to the 1 of my 2 picks (also chosen before the coin is ever flipped) that is closest. The one of us that is closest to the "real probability" wins $1,000,000.

What are your 2 picks? 0.5 & ???? If 0.5 is no longer one of your picks, how can it be the single best pick?

PairTheBoard

06-07-2007, 08:28 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I'd like to know if you now agree that the 0.5 pick is better than any other in the absence of other information about the coin and assuming indifference toward heads or tails.

[/ QUOTE ]

How can 0.5 be the best pick if it has a 0% confidence level (& a 0% chance of being correct, assuming a "bent coin" can't be bent in such a way as to make it a fair coin)?

[/ QUOTE ]

How about if you are allowed 2 picks prior to any flips of the coin. Then, after your 2 picks, the coin is flipped 1 billion times to find the probability that it will come up heads in any 1 flip. Whichever pick of yours is closest to this probability is then compared to the 1 of my 2 picks (also chosen before the coin is ever flipped) that is closest. The one of us that is closest to the "real probability" wins $1,000,000.

What are your 2 picks? 0.5 & ???? If 0.5 is no longer one of your picks, how can it be the single best pick?

[/ QUOTE ]

Trouble is, you could make that same argument even if you knew it was a perfectly fair coin. The best strategy for that game looks like it would be to make two symmetric picks super close to .5. One pair should optimize wrt the distribution for 1 million flips of a fair coin. That wouldn't change the fact that 50% is the probability the perfectly fair coin comes up heads on any specific flip, including the first one.

PairTheBoard

KipBond

06-07-2007, 08:47 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I'd like to know if you now agree that the 0.5 pick is better than any other in the absence of other information about the coin and assuming indifference toward heads or tails.

[/ QUOTE ]

How can 0.5 be the best pick if it has a 0% confidence level (& a 0% chance of being correct, assuming a "bent coin" can't be bent in such a way as to make it a fair coin)?

[/ QUOTE ]

How about if you are allowed 2 picks prior to any flips of the coin. Then, after your 2 picks, the coin is flipped 1 billion times to find the probability that it will come up heads in any 1 flip. Whichever pick of yours is closest to this probability is then compared to the 1 of my 2 picks (also chosen before the coin is ever flipped) that is closest. The one of us that is closest to the "real probability" wins $1,000,000.

What are your 2 picks? 0.5 & ???? If 0.5 is no longer one of your picks, how can it be the single best pick?

[/ QUOTE ]

Trouble is, you could make that same argument even if you knew it was a perfectly fair coin. The best strategy for that game looks like it would be to make two symmetric picks super close to .5. One pair should optimize wrt the distribution for 1 million flips of a fair coin. That wouldn't change the fact that 50% is the probability the perfectly fair coin comes up heads on any specific flip, including the first one.

[/ QUOTE ]

So, for a fair coin, you pick 0.499999 & 0.500001 -- Do you think those are good picks for the bent coin wager I proposed?

I suppose I could say 3 picks instead of 2. For a fair coin, 0.5 should still be one of your picks. For the bent coin, would/should it be?

PairTheBoard

06-07-2007, 10:11 PM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I'd like to know if you now agree that the 0.5 pick is better than any other in the absence of other information about the coin and assuming indifference toward heads or tails.

[/ QUOTE ]

How can 0.5 be the best pick if it has a 0% confidence level (& a 0% chance of being correct, assuming a "bent coin" can't be bent in such a way as to make it a fair coin)?

[/ QUOTE ]

How about if you are allowed 2 picks prior to any flips of the coin. Then, after your 2 picks, the coin is flipped 1 billion times to find the probability that it will come up heads in any 1 flip. Whichever pick of yours is closest to this probability is then compared to the 1 of my 2 picks (also chosen before the coin is ever flipped) that is closest. The one of us that is closest to the "real probability" wins $1,000,000.

What are your 2 picks? 0.5 & ???? If 0.5 is no longer one of your picks, how can it be the single best pick?

[/ QUOTE ]

Trouble is, you could make that same argument even if you knew it was a perfectly fair coin. The best strategy for that game looks like it would be to make two symmetric picks super close to .5. One pair should optimize wrt the distribution for 1 million flips of a fair coin. That wouldn't change the fact that 50% is the probability the perfectly fair coin comes up heads on any specific flip, including the first one.

[/ QUOTE ]

So, for a fair coin, you pick 0.499999 & 0.500001 -- Do you think those are good picks for the bent coin wager I proposed?

I suppose I could say 3 picks instead of 2. For a fair coin, 0.5 should still be one of your picks. For the bent coin, would/should it be?

[/ QUOTE ]

Actually, I suspect if you figured it out you could compute the optimum pair for the perfectly fair coin, and I suspect the pair would stradle 0.5 but be very close to it. A pair that included 0.5 would not optimize. So you could stick to a pair I think if you wanted. If you went to a triple then I think the optimizer would include 0.5.

If you use a pair for the bent coin, then I think phil would bring into play his second assumption about the prior distribution. His first assumption is that it is symmetric. His second assumption - when forced to it - is to assume something like 1/(x(1-x) with a hump in the middle. I don't know how good his hunch is about the shape of the humped distribution to give a guess for an optimum pair. For a Triple however, his symetric hump in the middle distribution would include 0.5 in the Triple. He would have to apply his second assumption to make that conclusion though. Symmetric distributions with a shallow center and large humps at both ends would make triples like (.1,.2,.85) look more attractive.

This is why I hesitate to tell him 0.5 is the "best" estimate for P(Heads on First Flip). It may be you can do no better. But suppose you wanted to measure "best" in a way that gave extra rewards to being close and didn't punish distant errors so much? That's essentially what you are proposing to do with this gambling application. The trouble is, it's not a gambling application that only depends on the First Flip of the Coin. If you could come up with something like this that only relied on the First Flip then you'd have something especially interesting. I'm not sure it's possible though.

The measure I was thinking about was averaging the square root of the absolute difference between your Guess and random Bending biases. I'm not sure it would make much sense to do that. But it might be a useful measure for some purposes. If that was the measure of "best" you used, then assuming symmetry would not be enough. For example, against a symmetric prior with extreme point masses of 50% at both 0 and 1, the guess of 0.5 would yield an average sqrt diff of about 0.7 while guesses of either 0 or 1 would both yield average sqrt differences of 0.5. The measure would "Reward" getting extra close.

PairTheBoard

jason1990

06-07-2007, 10:17 PM

[ QUOTE ]
His second assumption - when forced to it - is to assume something like 1/(x(1-x) with a hump in the middle.

[/ QUOTE ]
Peer review: The function f(x) = 1/(x(1 - x)) is convex with asymptotes at both endpoints.

PairTheBoard

06-07-2007, 10:33 PM

[ QUOTE ]
[ QUOTE ]
His second assumption - when forced to it - is to assume something like 1/(x(1-x) with a hump in the middle.

[/ QUOTE ]
Peer review: The function f(x) = 1/(x(1 - x)) is convex with asymptotes at both endpoints.

[/ QUOTE ]

Ah. Thanks. I saw zeroes but didn't notice they happened to be in the denominator. How about 6(x)(1-x). Is there something simple I can actually remember correctly?

PairTheBoard

jason1990

06-07-2007, 10:38 PM

It may be worth mentioning, in case you missed it, that this function is not integrable. So it is an improper prior.

KipBond

06-07-2007, 10:39 PM

[ QUOTE ]
If you went to a triple then I think the optimizer would include 0.5. ... For a Triple however, his symetric hump in the middle distribution would include 0.5 in the Triple. ... Symmetric distributions with a shallow center and large humps at both ends would make triples like (.1,.2,.85) look more attractive.

[/ QUOTE ]

I'm not sure what Phil would say, but it definitely seems to me that if he thinks his "best" guess is 0.5, then he would definitely have to include that as one of his 3 picks (as you say). I think your 3 picks (.1, .2, .85) could be better picks than his. I was thinking .2, .3, .75 when I proposed the triple-pick question, actually. /images/graemlins/smile.gif

chezlaw

06-07-2007, 10:40 PM

[ QUOTE ]
Notice he did not say, "Two people can have two different assessments of a situation even though they have the same information." I am pretty sure he disagrees with that sentence. After all, he does not believe that probability is subjective, so probability cannot depend on the person, only on the information (according to his words, as I understand them).

[/ QUOTE ]
I'm not sure this is a problem for DS. If given the information there's no rational way to tell which assessment is better then there's no advantage to either assessment when making a decision.

So I'm a bit confused. If you were saying there are cases where someone can use their assessment to make better decisions than other perfectly rational people who have exactly the same information then that would be a problem but that can't be right, can it?.

chez

m_the0ry

06-08-2007, 12:43 AM

This thread is getting really confusing. Where did the probability distributions come in? Coin flips are ideally modeled by a binomial discrete probability which always has the form

(n x)*(p^x)*(1-p)^(n-x)

and so the best 'picks' depend on n, the number of trials, and nothing else. Because it's discrete the best picks are always the *possible* sample means closest to .5 . for example if I have 2 picks out of 1 million trials I always pick .5 as the sample mean, and then 499999 vs 500001 as the next best sample mean. The symmetry of the problem gives that the name associated with the 499999 figure and 500001 figures are meaningless.

jason1990

06-08-2007, 12:54 AM

[ QUOTE ]
[ QUOTE ]
Notice he did not say, "Two people can have two different assessments of a situation even though they have the same information." I am pretty sure he disagrees with that sentence. After all, he does not believe that probability is subjective, so probability cannot depend on the person, only on the information (according to his words, as I understand them).

[/ QUOTE ]
I'm not sure this is a problem for DS. If given the information there's no rational way to tell which assessment is better then there's no advantage to either assessment when making a decision.

[/ QUOTE ]
Here is where David defines "probability":

[ QUOTE ]
Anyway before someone reasks me the question which I will answer, let me say again what I mean by "probability".

Probability is related to the information you have about a subject. Two people can have two different assessments of a situation because they have different information. There is no "right" answer. Is the next card turned going to be the ace of spades. You will give a probability estimate based on what cards you have seen. Even a fair coin is not even money when flipped if you have some physical information about the flipper.

When the information is not as oviously correlated with probability as seen cards, you resort to your knowledge of that information. A black and a white horse are about to race. You know nothing else about them. If you happen to have knowledge of other races involving black and white horses, say the black horse won 45,000 out of 100,000 you should use that ratio.

[/ QUOTE ]
Okay, so this is not really a "definition". It is more like a description of what you should do in various situations if you want to compute probabilities. But it is clear to me that this says the "probability" of an event is determined by the information and knowledge in your possession. If two people have the exact same information and knowledge, then they must come up with the exact same "probability". If they do not, then they are using the word "probability" in a way which is different from David's.

chezlaw

06-08-2007, 02:26 AM

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Notice he did not say, "Two people can have two different assessments of a situation even though they have the same information." I am pretty sure he disagrees with that sentence. After all, he does not believe that probability is subjective, so probability cannot depend on the person, only on the information (according to his words, as I understand them).

[/ QUOTE ]
I'm not sure this is a problem for DS. If given the information there's no rational way to tell which assessment is better then there's no advantage to either assessment when making a decision.

[/ QUOTE ]
Here is where David defines "probability":

[ QUOTE ]
Anyway before someone reasks me the question which I will answer, let me say again what I mean by "probability".

Probability is related to the information you have about a subject. Two people can have two different assessments of a situation because they have different information. There is no "right" answer. Is the next card turned going to be the ace of spades. You will give a probability estimate based on what cards you have seen. Even a fair coin is not even money when flipped if you have some physical information about the flipper.

When the information is not as oviously correlated with probability as seen cards, you resort to your knowledge of that information. A black and a white horse are about to race. You know nothing else about them. If you happen to have knowledge of other races involving black and white horses, say the black horse won 45,000 out of 100,000 you should use that ratio.

[/ QUOTE ]
Okay, so this is not really a "definition". It is more like a description of what you should do in various situations if you want to compute probabilities. But it is clear to me that this says the "probability" of an event is determined by the information and knowledge in your possession. If two people have the exact same information and knowledge, then they must come up with the exact same "probability". If they do not, then they are using the word "probability" in a way which is different from David's.

[/ QUOTE ]
We're just going to have to leave it up to DS to clarify what he means (or not knowing DS)

[ QUOTE ]
If two people have the exact same information and knowledge, then they must come up with the exact same "probability". If they do not, then they are using the word "probability" in a way which is different from David's.

[/ QUOTE ]
That's the key bit. Coming up with the same probability may include NULL (not unknown but non-existent).

chez

David Sklansky

06-08-2007, 03:34 AM

His position, therefore, seems pretty clear to me: Information determines probability. Since "not enough information" is never a necessary conclusion, information always determines probability. Since the same information cannot produce two different probabilities, information determines a unique probability.

If you know the probability and you know the odds you are getting, you can always compute an EV and make your decision. So he is asserting (at least the existence of) an algorithm which, in theory, takes the information you have and produces a definitive decision. Not only that, this algorithm is logically superior to any other algorithm, since there is only one unique probability, and therefore only one unique decision, for any given set of information. He may not claim to know that algorithm in all its gory details. But he believes in its existence.

This is what that group of mathematicians would like to develop. But some other mathematicians think their efforts are in vain, because they believe such an algorithm does not exist. You could say they are the "algorithm atheists" and David is an "algorithm theist". There is no proof either way, since no such algorithm has been created and agreed upon, even in the simplest of cases.

I have not thought about this in your rigorous hi fallootin way. So I am an algorithm agnostic. But I think I see a possible way to disprove the algorithm theists. Create a paradox. Come up with a probability problem, perhaps one involving probability problems or probability professors. Assume there is an algorithm that will come up with a specific answer with the information given, and then add THAT ANSWER to the original information. If you can come up with a problem where the addition of that new information somehow changes the answer (ie if the answer is 30% then the answer is 35%) you have your Godel like proof that those mathmeticians are wasting their time.

jason1990

06-08-2007, 04:42 AM

[ QUOTE ]
I have not thought about this in your rigorous hi fallootin way. So I am an algorithm agnostic. But I think I see a possible way to disprove the algorithm theists. Create a paradox. Come up with a probability problem, perhaps one involving probability problems or probability professors. Assume there is an algorithm that will come up with a specific answer with the information given, and then add THAT ANSWER to the original information. If you can come up with a problem where the addition of that new information somehow changes the answer (ie if the answer is 30% then the answer is 35%) you have your Godel like proof that those mathmeticians are wasting their time.

[/ QUOTE ]
You should work on this. It could be a big breakthrough. Or at least a neat little article. Post it when you find it.

TomCowley

06-08-2007, 01:08 PM

The 2 and 3 guesses problems really has nothing to do with a best estimate of P. I've had real-life experience with the two-guesses case- I did a 3-week analytical chemistry lab experiment that used two methods to determine the same number, and when I was going to do the final calculations... realized I hadn't written down how much of a chemical I'd started with. At this point, all I knew was that I started with between 0.7g and 0.8g. The experiment was graded on the closest of the two numbers. All I could do was trust that my work was accurate, and then figure out the proper guesses for starting material to maximize my grade. Even though my clear best single guess would be starting with ~0.75g, the actual optimal guesses were closer to .725 and .775 to best cover the entire range. Long story short- I got a 97.

PairTheBoard

06-08-2007, 05:00 PM

I think you might like this phil. I just made this post on David's Newer Thread about the "Question" he hadn't answered. Link (http://forumserver.twoplustwo.com/showthreaded.php?Cat=0&Number=10715757&page=0&vc=1 )

It speaks to your criticism of jason1990's original statement on this thread, so jason might be interested too. It's a reply to a KipBond post in which I thought Kip was making an unfair criticism of Sklansky's statement of probability about the first flip of the Bent Coin.

I think it also sheds light on what Jason is doing with his "Purely Abstract Problem" thread. He is examining the Procedure that Sklansky uses. As I point out below, in this case the Procedure is much more useful that the initial result it provides for the probability of heads for First Flip of Bent Coin. This is also why I kept stressing my need to look at your Procedure for arriving at that initial assesment. The Procedure is more useful than it's initial application.

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[ QUOTE ]
KipBond -
Surely with a fair coin this is extremely simple. The P(H)=50%, so you pick the range: [475,000 - 525,000]. You expect to win a bunch of money over the 10M trials.

If you are sure about your P(Heads of Bent Coin)=50%, then why wouldn't you pick the same range as you would for the fair coin wager -- the one that is guaranteeing you many millions of dollars? Why aren't you using your probability to make the best decision? Why, instead, are you insisting on randomizing your decision?

[/ QUOTE ]

The thing to realize here is that Sklansky's Baysian definition of Probability is not the same as the Frequentist. The Frequentist is the one on which mathematical models of probability are based. That doesn't mean that mathematical probability models can't be applied by Baysians. It does mean we have to treat the two definitions differently. It is not a fair criticism of a Baysian statement of probability to treat it as if a Frequentist had make it.

On David's Thread in the Probablity Forum, jason1990 just provided a link to an explanation by Persi Diaconis of a Baysian definition of probilility. I wonder if Sklansky disagrees with any of it. Basically I think Persi provides some of the extra details and rigor to the definition that illuminates what Sklansky defines as Probability. This is the material we should be looking at to see if Sklansky stays consistent in how he makes use of Probability according to his definition. There's no point in making unfair attacks on his definition.

Persi Diaconis on Subjective Probability (http://www-stat.stanford.edu/~cgates/PERSI/courses/stat_121/lectures/subjectiveprob/)

So when you say,

[ QUOTE ]
If you are sure about your P(Heads of Bent Coin)=50%, then why wouldn't you pick the same range as you would for the fair coin wager

[/ QUOTE ]

you are making an unfair attack. Notice Sklansky did not agree to the proposition,

"P(Heads of Bent Coin)=50% (Not a Sklansky statement)"

He agreed to the proposition,

"P(Heads on first flip of Bent Coin)=50% (Sklansky statement)"

Notice I made the conditions, (Not a Sklansky Statement) and (Sklansky Statement) part of the propositions. You might say you were just lazy about writing the "on the first flip" part because that really doesn't matter to you. But it matters to the Baysian and it matters to Sklansky's definition of probability. To make this perfectly clear, Sklansky did NOT agree to the following proposition,

"P(head on first flip of Bent Coin)= 50% (Frequentist statement)"

Sklansky did Not agree to that proposition. So it's not fair to criticize the proposition he agreed to by treating it as if it were a proposition he did not agree to.

If you read the Persi Diaconis link you will see that the requirement he puts on the Baysian Definition of probability is that it not produce any "Dutch Books". He cannot agree to two probability statements whereby his indifference to the odds they would imply for gambling propositions would allow you to make a Dutch Book on him and win money betting both sides of both propositions and automatically win money because of their inconsistency. For example, Sklansky could not say,

"The 6 sides of the Mystery loaded Die are all equally likely on the First roll"

and then also say,

"P(Mystery loaded die comes up 3 on First Roll) = 1/8".

He cannot agree to both those statements because a gambler could make a Dutch Book on him knowing nothing about how the die is loaded.

So when David tries to determine his answer to your gambling proposition with respect to his assertion,

"P(Heads on first flip of Bent Coin)=50% (Sklansky statement)"

he should be doing so in such a way that a Dutch Book cannot be made on him between his statement above and what he says about your gambling proposition. If you think he is failing to do that you have an argument.

But remember, the only other gambling proposition he has said his assertion implies is that he is indifferent to betting heads or tails on an even money first flip of the bent coin. Being indifferent the only way he sees to decide which to bet on IF FORCED, is to flip a fair coin. Something I do myself sometimes in undecidable situations - although I often second guess the flip.

What that implies about gambling propositions that involve flipping the coin more than once is unclear. It may imply nothing, which makes us wonder about its usefulness. But then again that might be ok. The Usefulness comes from the Procedure that went into making that Baysian probability statement. I think we can interrogate him on whether he is using that Procedure consistently when he decides what to say about your new gambling proposition.

We would also like to know more about exactly what that Procedure is. How easy is it to apply? If we apply it once how can we use the result? Or is the result an isolated thing which we can't build on. Do we have to start all over again each time we get a new problem? Is there any way of incorporating the results of one application of the Procedure into the Procedure we use in the next application?

Your new Gambling Proposition is shedding some light on those questions. You are pointing out how the Frequentist assertion,

"P(head on first flip of Bent Coin)= 50% (Frequentist statement)"

would be easy to build on and apply to your new Gambling Proposition. While Sklansky's assertion looks to be one that can't be built on and applied. He must start from scratch so to speak and apply the Procedure that went into his assertion all over again to respond to your new Gambling Proposition. However, this is not a fair criticism either in this case because the Frequentist assertion is almost certainly False. We don't have that assertion from the Frequentist here. The statement we do have from the Frequentist is,

"P(head on first flip of Bent Coin)= not enough information (Frequentist statement)"

We hardly have a case that we can build much on that statement either. At least Sklansky's statement provides a way for us to gamble on the first flip of the Bent Coin.
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PairTheBoard