Clarifying The Bent Coin Problem

[chezlaw]

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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).

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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.

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Here is where David defines "probability":

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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.

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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.

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

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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.

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

chez

[David Sklansky]

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]

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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.

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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]

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]

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

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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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?


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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


So when you say,

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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

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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

[DiceyPlay]

11-10 on a fair coin - take the bet over and over and become a millionaire. The only obstacle is bankroll management - you don't want to go broke on a run of bad luck.

Bent coin - take the bet over and over. The bias will manifest itself after a certain number of flips (that number of flips depends on how strong the bias is). Bankroll management still applies. You must still monitor the situation as it progresses (review observations and adjust if necessary).

In the first scenario the bet has a bias (value of a bet concept) and you have a positive expectation - it actually doesn't matter if you choose heads or tails on any flip.

In the second scenario the flip has a bias. You must determine the bias and the bias will manifest itself as you observe outcomes of flips.

[Munche]

As long as you are truly random in your guess, and the same bent coin is used regardless of your guess, take the bent coin bet. You negate all the bet offerers plans and make the bent coin fair with better odds. A 100% loaded coin, if you choose randomly, could actually be fairer than the "fair" coin as chances are with enough work someone could toss a fair coin to whichever side they wanted more than 50% of the time.