No Universal Algorithm Regarding Information/Probability Can Exist

[David Sklansky]

Suppose there was such an algorithm which shows you how to translate your information into a probability. El Diablo studies it and posts that I have a 25% chance to have sex with Brandi. Obviously when I read this I want to prove him wrong. Thereby increasing the chances.

Notice that it isn't necessary to bring someone else into the picture. I could have plugged all the information into the algorithm myself. And become pissed at the answer.

Notice also that it does no good to include into the original infrormation the fact that I would be reading its prediction. That new information would probably adjust the probability prediction but it still can't escape the fact that I have a prediliction to proving algorithms wrong. And even if it knows THAT, its screwed. Sort of like how Cantor's diagonalisation proof that real numbers are uncountable works.

[haydenpan]

I don't get it, but you're awesome.

[chezlaw]

Have you disproved determinism or just proved that even if the world is determinisic it can't be predicted?

chez

[Duke]

What you're saying is equivalent to saying that you can never have perfect information. Your OP doesn't offer anything to distinguish itself from that idea, which is already known.

We're back to you saying that you can't have a universal algorithm, and me saying "sure you can, you just need perfect information." You give your example, and I say that that's exactly the same thing as additional information.

[SNOWBALL]

This OP reads like the first section of <u>Notes From Undeground</u> by Dostoyevsky.

[Kimbell175113]

I'm sure that every time C-3PO told Han Solo the odds of something, that only made the ol' smuggler try harder.

Now you're saying it also made Threepio wrong? Whoa, I don't know if I like this theory.

[Metric]

I think a concrete realization of this would be a program that downloads all relevant information from the internet, does some clever manipulation, and then predicts the price of stock for the next quarter, with incredible accuracy (or in terms of perfectly accurate probabilities).

It's pretty easy to see that if I sold this program to the general public for $19.99 plus tax, it would become far less useful in a very short period of time. Why? Because stock values would immediately change based on the predictions of the program.

[Duke]

[ QUOTE ]
I think a concrete realization of this would be a program that downloads all relevant information from the internet, does some clever manipulation, and then predicts the price of stock for the next quarter, with incredible accuracy (or in terms of perfectly accurate probabilities).

It's pretty easy to see that if I sold this program to the general public for $19.99 plus tax, it would become far less useful in a very short period of time. Why? Because stock values would immediately change based on the predictions of the program.

[/ QUOTE ]

So probabilities can change over time based on new information? The hell you say.

I still see this whole "concept" as nothing more than a pseudo-formalization of the idea that it's not really possible to have perfect information. Data about data can be new data as well - there's no practical difference in this context between metadata (David learning about the odds and working to change them) and data (whatever set of information that did not include that that diablo based his original assessment on).

[FortunaMaximus]

Maxwell's Demon.

Claude Shannon's Demon.

Sklansky's Demoness?

Exploitability is underrated.

[Duke]

[ QUOTE ]
I think a concrete realization of this would be a program that downloads all relevant information from the internet, does some clever manipulation, and then predicts the price of stock for the next quarter, with incredible accuracy (or in terms of perfectly accurate probabilities).

It's pretty easy to see that if I sold this program to the general public for $19.99 plus tax, it would become far less useful in a very short period of time. Why? Because stock values would immediately change based on the predictions of the program.

[/ QUOTE ]

It's also pretty easy to see that you could use that information to create a new program to sell for $29.95 a pop that would take that into consideration. And then $39.95 on that.

Yeah, the pyramid grows forever, but it's still just new information.

[felson]

This argument depends on a high probability of the simultaneous occurrence of four events:

- DS hears/calculates the "correct" probability prediction
- DS is motivated to disprove the prediction
- DS is consistently motivated to disprove in the same direction ("try harder" vs "give up")
- DS is capable of nontrivially altering the course of events as desired

I don't see why it would be highly likely that all four would occur at once.

[Metric]

[ QUOTE ]
[ QUOTE ]
I think a concrete realization of this would be a program that downloads all relevant information from the internet, does some clever manipulation, and then predicts the price of stock for the next quarter, with incredible accuracy (or in terms of perfectly accurate probabilities).

It's pretty easy to see that if I sold this program to the general public for $19.99 plus tax, it would become far less useful in a very short period of time. Why? Because stock values would immediately change based on the predictions of the program.

[/ QUOTE ]

It's also pretty easy to see that you could use that information to create a new program to sell for $29.95 a pop that would take that into consideration. And then $39.95 on that.

Yeah, the pyramid grows forever, but it's still just new information.

[/ QUOTE ]
Sure. I think David's point (phrased in terms of this example) was simply that there can never be a final version of this software that does it all and can't be improved by exactly this sort of iterative process. Maybe it's obvious, but it does seem like it should be important.

[PairTheBoard]

The hypothetical algorithm produces a perfectly processed information based credence for the Event. It doesn't produce a perfect infomation based probability for the event unless it has perfect information. If its information includes your prediliction for responding to its credence results about you then it can preadjust its output credence, sliding it up or down the proper amounts. As it slides the credence output up you may take less offense to it. At some point it will balance, having accounted for whatever extra efforts you will be making.

So no. I think Duke is right. This doesn't provide any kind of theoretical proof of anything. If the infomation the algorithm is processing doesn't include your predilicions then the credence output will be flawed because of the lack of information, not because the algorithm didn't process its information input perfectly.

PairTheBoard

[David Sklansky]

[ QUOTE ]
The hypothetical algorithm produces a perfectly processed information based credence for the Event. It doesn't produce a perfect infomation based probability for the event unless it has perfect information. If its information includes your prediliction for responding to its credence results about you then it can preadjust its output credence, sliding it up or down the proper amounts. As it slides the credence output up you may take less offense to it. At some point it will balance, having accounted for whatever extra efforts you are making.

So no. I think Duke is right. This doesn't provide any kind of theoretical proof of anything. If the infomation the algorithm is processing doesn't include your predilicions then the credence output will be flawed because of the lack of information, not because the algorithm didn't process its information input perfectly.

PairTheBoard

[/ QUOTE ]

But what about if my predilections are to not accept algorithmic predictions about me?

[bluesbassman]

Mr. Sklansky's argument is similar to the ideas contained in Godel's celebrated first incompleteness theorem (http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem). From wiki here is a non-technical statement of that theorem:

For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

This theorem essentially states that it is impossible to construct a "universal algorithm" which can generate all true arithmetic expressions of number theory. The proof demonstrates it is always possible to "jump out" of a particular formal system and construct a true statement which is not provable in that system. Note that these formal systems are deterministic, not probabilistic.

A wonderful book which illuminates Godel's theorem in a very entertaining manner, as well as discusses many other fascinating ideas, is Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid (http://en.wikipedia.org/wiki/G%C3%B6del%2C_Escher%2C_Bach).

[jogger08152]

[ QUOTE ]
[ QUOTE ]
The hypothetical algorithm produces a perfectly processed information based credence for the Event. It doesn't produce a perfect infomation based probability for the event unless it has perfect information. If its information includes your prediliction for responding to its credence results about you then it can preadjust its output credence, sliding it up or down the proper amounts. As it slides the credence output up you may take less offense to it. At some point it will balance, having accounted for whatever extra efforts you are making.

So no. I think Duke is right. This doesn't provide any kind of theoretical proof of anything. If the infomation the algorithm is processing doesn't include your predilicions then the credence output will be flawed because of the lack of information, not because the algorithm didn't process its information input perfectly.

PairTheBoard

[/ QUOTE ]

But what about if my predilections are to not accept algorithmic predictions about me?

[/ QUOTE ]
The stronger your predilection is, the closer to 1/2 the algorithm will predict.

[felson]

[ QUOTE ]
Mr. Sklansky's argument is similar to the ideas contained in Godel's celebrated first incompleteness theorem (http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem). From wiki here is a non-technical statement of that theorem:

For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

This theorem essentially states that it is impossible to construct a "universal algorithm" which can generate all true arithmetic expressions of number theory. The proof demonstrates it is always possible to "jump out" of a particular formal system and construct a true statement which is not provable in that system. Note that these formal systems are deterministic, not probabilistic.

A wonderful book which illuminates Godel's theorem in a very entertaining manner, as well as discusses many other fascinating ideas, is Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid (http://en.wikipedia.org/wiki/G%C3%B6del%2C_Escher%2C_Bach).

[/ QUOTE ]

Like Godel, Sklansky's argument essentially boils down to the paradox "this sentence is false." The difference is that Sklansky's argument is not airtight, as I pointed out a few posts ago.

[Arp220]

Hello, new to these forums, and this looked like an interesting question /images/graemlins/wink.gif

ok:

If such an algorithm were to exist, then it would be sufficiently complex that any specific numerical prediction it made would be conditional on certain circumstances. For example, this El Diablo person would compute a 25% chance that you will have sex with Brandi (I hope she's hot), but that specific computations reliability would likely depend on you not being aware of the computation being done, or more likely that the algorithm even exists. If you are aware of either thing, then a 'universal' algorithm would factor that in and revise its computation for that circumstance.

However, this whole problem does sound eerily reminiscent of Godels incompleteness theorem....

[Metric]

[ QUOTE ]
However, this whole problem does sound eerily reminiscent of Godels incompleteness theorem....

[/ QUOTE ]
Yeah, that and the halting problem -- it's the self-referential nature of the thing.

[superadvisor]

My take on the Universal Algorithm is that it could exist but it's output could only measure probability in an instant. Information is always changing and in order to keep up with the constantly changing information, every time new information comes in to existence, such as David seeing the score and reacting to it, a new cycle of the algorithm must be executed to take in to account the new information. Even so, the output as I said before would only be accurate for the instant it was processed, moments later the probability most likely would be very different.

[David Sklansky]

Just to make it clear, the only reason I wrote or thought about this was because Jason said a bunch of mathmeticians are trying to find such a universal algorithm.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
The hypothetical algorithm produces a perfectly processed information based credence for the Event. It doesn't produce a perfect infomation based probability for the event unless it has perfect information. If its information includes your prediliction for responding to its credence results about you then it can preadjust its output credence, sliding it up or down the proper amounts. As it slides the credence output up you may take less offense to it. At some point it will balance, having accounted for whatever extra efforts you are making.

So no. I think Duke is right. This doesn't provide any kind of theoretical proof of anything. If the infomation the algorithm is processing doesn't include your predilicions then the credence output will be flawed because of the lack of information, not because the algorithm didn't process its information input perfectly.

PairTheBoard

[/ QUOTE ]

But what about if my predilections are to not accept algorithmic predictions about me?

[/ QUOTE ]

That's the kind of predilections I was talking about. What about it?

PairTheBoard

[PairTheBoard]

[ QUOTE ]
Mr. Sklansky's argument is similar to the ideas contained in Godel's celebrated first incompleteness theorem (http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorem). From wiki here is a non-technical statement of that theorem:

For any consistent formal, computably enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true but not provable in the theory can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

This theorem essentially states that it is impossible to construct a "universal algorithm" which can generate all true arithmetic expressions of number theory. The proof demonstrates it is always possible to "jump out" of a particular formal system and construct a true statement which is not provable in that system. Note that these formal systems are deterministic, not probabilistic.

A wonderful book which illuminates Godel's theorem in a very entertaining manner, as well as discusses many other fascinating ideas, is Hofstadter's Godel, Escher, Bach: An Eternal Golden Braid (http://en.wikipedia.org/wiki/G%C3%B6del%2C_Escher%2C_Bach).

[/ QUOTE ]

I don't think that's exactly what it says. It does say there are such "jump out" statements. But it does not say they are "True" statements. They are statements which are neither True nor False with respect to the implications of the axioms for the system. They can equally well be assumed True or False. For example, if P is such a statement, then either P or not-P can be added to the axioms of the system without producing any inconsistencies to the new system of axioms.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
Just to make it clear, the only reason I wrote or thought about this was because Jason said a bunch of mathmeticians are trying to find such a universal algorithm.

[/ QUOTE ]

I don't think he said they were trying to find one. Only to prove whether or not one could theoretically exist. If they proved it one way or the other it could be used in formulating theories of probability.

PairTheBoard

[David Sklansky]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
The hypothetical algorithm produces a perfectly processed information based credence for the Event. It doesn't produce a perfect infomation based probability for the event unless it has perfect information. If its information includes your prediliction for responding to its credence results about you then it can preadjust its output credence, sliding it up or down the proper amounts. As it slides the credence output up you may take less offense to it. At some point it will balance, having accounted for whatever extra efforts you are making.

So no. I think Duke is right. This doesn't provide any kind of theoretical proof of anything. If the infomation the algorithm is processing doesn't include your predilicions then the credence output will be flawed because of the lack of information, not because the algorithm didn't process its information input perfectly.

PairTheBoard

[/ QUOTE ]

But what about if my predilections are to not accept algorithmic predictions about me?

[/ QUOTE ]

That's the kind of predilections I was talking about. What about it?

PairTheBoard

[/ QUOTE ]

Your original post pointed out that if I change my effort base on being insulted by lack of faith in me, there will be an equilibrium point. But if I am insulted by the mere fact that an algorithm purpots to be able to predict me, there is no equilibrium.

[David Sklansky]

[ QUOTE ]
[ QUOTE ]
Just to make it clear, the only reason I wrote or thought about this was because Jason said a bunch of mathmeticians are trying to find such a universal algorithm.

[/ QUOTE ]

I don't think he said they were trying to find one. Only to prove whether or not one could theoretically exist. If they proved it one way or the other it could be used in formulating theories of probability.

PairTheBoard

[/ QUOTE ]

Obviously. Meanwhile what do you think of the idea that the Brandi-Sklansky Paradox might hammer a stake in the heart of Bayesians?

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
The hypothetical algorithm produces a perfectly processed information based credence for the Event. It doesn't produce a perfect infomation based probability for the event unless it has perfect information. If its information includes your prediliction for responding to its credence results about you then it can preadjust its output credence, sliding it up or down the proper amounts. As it slides the credence output up you may take less offense to it. At some point it will balance, having accounted for whatever extra efforts you are making.

So no. I think Duke is right. This doesn't provide any kind of theoretical proof of anything. If the infomation the algorithm is processing doesn't include your predilicions then the credence output will be flawed because of the lack of information, not because the algorithm didn't process its information input perfectly.

PairTheBoard

[/ QUOTE ]

But what about if my predilections are to not accept algorithmic predictions about me?

[/ QUOTE ]

That's the kind of predilections I was talking about. What about it?

PairTheBoard

[/ QUOTE ]

Your original post pointed out that if I change my effort base on being insulted by lack of faith in me, there will be an equilibrium point. But if I am insulted by the mere fact that an algorithm purpots to be able to predict me, there is no equilibrium.

[/ QUOTE ]

That will all be factored in if the prediliction information is avaialable. At some point you will be insulted at a level where your extra effort produces a probability of success that matches the level. If your prediliction is such that the balance level is 100% then that's the output the algorithm will produce. But that doesn't seem likely. 100% will not insult you. In fact that might make you put in less effort. So the level would have to be less than that. Some level would produce a balance though. The algorithm will find it if it has the correct prediliction/level information.

I suppose it's possible that such balance levels are not unique. I'm not sure what that would imply. Maybe if there are two such levels the algorithm would apply Sklansky's Principle and say the two probabilities are equally likely. I'm not sure what that would mean though. I guess we would flip a coin to decide what odds to take on the bet.

This is geting too high falootin for me.

PairTheBoard

[Maple Leafs]

[ QUOTE ]
El Diablo studies it and posts that I have a 25% chance to have sex with Brandi.

[/ QUOTE ]
Brandi is a woman. She's knew the exact odds of having sex with you the moment she met you, and there is no action you could take or not take that will change that percentage.

[fraac]

How do you prove a percentage wrong? (Apart from 0%)

Also, anyone saying you're wrong is right.

[ruken]

[ QUOTE ]
[ QUOTE ]
El Diablo studies it and posts that I have a 25% chance to have sex with Brandi.

[/ QUOTE ]
Brandi is a woman. She's knew the exact odds of having sex with you the moment she met you, and there is no action you could take or not take that will change that percentage unless unexpected information comes up implying that you are significantly more or less wealthy than she had initially assumed, in which case the chances of having sex with you are significantly adjusted in kind.

[/ QUOTE ]

FYP

[fraac]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
El Diablo studies it and posts that I have a 25% chance to have sex with Brandi.

[/ QUOTE ]
Brandi is a woman. She's knew the exact odds of having sex with you the moment she met you, and there is no action you could take or not take that will change that percentage unless unexpected information comes up implying that you are significantly more or less wealthy than she had initially assumed, in which case the chances of having sex with you are significantly adjusted in kind.

[/ QUOTE ]

FYP

[/ QUOTE ]
Brandi is a crazy woman. She knows what she needs to know to get through the next ten minutes, the next room of people. She could be convinced you have not a hope in hell with her, then you do something unexpected and you've become a totally different person in her eyes.

There you go, Sklanners. Threw you a bone.

[LeadbellyDan]

Isn't this similar to the self halting problem?

First number all the Turing machines so that each machine is given a number, e. Now consider the set K of numbers e such that the eth Turing machine halts when given input e (its own number).

This set is not recursive - there is no Turing machine which when given input e gives output 1 if e is a member of K and 0 if it is not.

To show this assume there is some Turing machine which does do this, H. Now consider the slightly larger machine which comprises H plus an infinite loop when H's output is 1 and nothing otherwise.

If this is the kth Turing machine then the kth machine halts iff H says it doesnt halt. Therefore H does compute the members of K.

In your example you take the place of the infinite loop when you look at the output of the probability algorithm. Whatever the output, you can change it, for example by making it zero.

I dont think these arguments are actually diagonalizations but they do have the self referential elements of diagonalization.

[borisp]

[ QUOTE ]
Suppose there was such an algorithm which shows you how to translate your information into a probability...but it still can't escape the fact that I have a prediliction to proving algorithms wrong.

[/ QUOTE ]
Except it can, because it's an algorithm. Your failure of imagination has identified at what point you need to resort to a contradiction to prove your point, which means you need to keep trying.

[ QUOTE ]
And even if it knows THAT, its screwed. Sort of like how Cantor's diagonalisation proof that real numbers are uncountable works.

[/ QUOTE ] No, because Cantor's argument actually constructs the real number that is needed to prove that there is no surjective map from the integers to the interval [0,1]. It relies on the principle of induction, which is an axiom, to do this. You, on the other hand, throw up your arms in disgust and say "I can't think any more, it must be true!!"

The construction necessary for this argument, that there is no universal measure, appeals to the axiom of choice as far as I know, in the purely set theoretical sense. And if you interpret it reasonably as stated, then any nonmeasurable set (see "Banach Tarski paradox") is a legitimate justification of its truth in a very broad sense.

Any serious mathematician would very significantly qualify any legitimate attempt at something along the lines of what the alleged "Jason" was talking about. Anyone who doesn't qualify it heavily does not know of what they speak.

[Borodog]

David,

I am sorry that I missed this post when you originally made it, thankfully, somebody pointed it out and called it a fallacy.

If I am reading you correctly, it is emphatically NOT a fallacy; it is something that the Austrian economists have been saying for decades (sans the Brandi sex examples).

[TomCowley]

This isn't even really about "probability". It reduces to a much more fundamental paradox based on three assumptions.

1) The future is deterministic (or in a probabilistic/quantum sense, all future states and their likelihoods are known).

2) A person is aware of some aspect of the deterministic prediction that his actions can influence.

3) A person has free will.

Three guesses which assumption is the worst.

[mrick]

[ QUOTE ]
David,

I am sorry that I missed this post when you originally made it, thankfully, somebody pointed it out and called it a fallacy.

If I am reading you correctly, it is emphatically NOT a fallacy; it is something that the Austrian economists have been saying for decades.

[/ QUOTE ]I bet you dont even agree with Sklansky or borisp about the definition of probability... /images/graemlins/smile.gif

WHAT DOES THE TERM PROBABILITY MEAN ?? I cant say.

[Dale Dough]

You still haven't slept with Brandi??

[m_the0ry]

[ QUOTE ]
This isn't even really about "probability". It reduces to a much more fundamental paradox based on three assumptions.

1) The future is deterministic (or in a probabilistic/quantum sense, all future states and their likelihoods are known).

2) A person is aware of some aspect of the deterministic prediction that his actions can influence.

3) A person has free will.

Three guesses which assumption is the worst.

[/ QUOTE ]

QFT, this OP is a little ridiculous.

[jason1990]

The thing that Sklansky is trying to disprove with this OP does not really have anything to do with determinism or free will or any of that. The idea that Sklansky is reacting to is inconveniently absent from this thread, so allow me to correct that by presenting it here.

There is a certain school of thought called logical probability which regards probabilities as extended truth values -- 0 is false, 1 is true, and anything in between is a "degree of plausibility." In logical probability, the laws of probability are simply the rules of inference that apply to these extended truth values. In logical probability, any two rational persons with the same information should arrive at the same probability assignments. So, given a particular set of information regarding some proposition, there is a "correct" probability that should be assigned to it.

For example, suppose that I know that Sklansky is about to play a game in which he is either going to win or lose. Suppose I genuinely know nothing else about Sklansky or the game. Based on this information, what is the probability Sklansky will win? According to logical probability, the correct answer is 1/2. This comes from the indifference principle, which is a fundamental part of logical probability. If I announce my conclusion to Sklansky and he reacts by adjusting his playing style in the game, then that is completely irrelevant. Nothing Sklansky does changes the fact that, based on the information I had, the correct probability assignment was 1/2. The correctness of this probability comes from the indifference principle, which is an axiom of sorts in logical probability, and which loosely asserts that no information corresponds to equal probabilities. In other words, the correctness of this probability is a property of the information, not of the game itself. Observing the actual game will neither support nor refute the validity of this probability assignment.

In more complicated examples, it may be nearly impossible to convert information into probability. For instance, based on all the information you presently have, what is the probability that the U.S. military will leave Iraq before December 31, 2008? According to logical probability, the information you presently have, since it is some fixed set of information, can correspond to at most one probability assignment for this proposition. There is no room for opinion. The probability is a function of the information. The "correct" probability is the one that a perfectly rational person with your information would arrive at. The "universal algorithm" to which Sklansky refers would simply be a complete set of rules that would, ideally, carve out a path from the given set of information to this perfectly rational probability assignment.

[borisp]

[ QUOTE ]
There is a certain school of thought called logical probability which regards probabilities as extended truth values -- 0 is false, 1 is true, and anything in between is a "degree of plausibility."

[/ QUOTE ]
This "school of thought" is nonsense. However, to prove that it is nonsense, you cannot use nonsense.

[jason1990]

[ QUOTE ]
[ QUOTE ]
There is a certain school of thought called logical probability which regards probabilities as extended truth values -- 0 is false, 1 is true, and anything in between is a "degree of plausibility."

[/ QUOTE ]
This "school of thought" is nonsense. However, to prove that it is nonsense, you cannot use nonsense.

[/ QUOTE ]
I may end up agreeing with you, but I am presently trying to remain open-minded. As far as I can tell, logical probability is alive and kicking, as far as philosophical interpretations of probability are concerned. I doubt that you, as someone who does mathematics and not philosophy, could unequivocally prove that an entire branch of philosophy is nonsense. However, if you would like to try, I might be interested to read what you have to write.

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
There is a certain school of thought called logical probability which regards probabilities as extended truth values -- 0 is false, 1 is true, and anything in between is a "degree of plausibility."

[/ QUOTE ]
This "school of thought" is nonsense. However, to prove that it is nonsense, you cannot use nonsense.

[/ QUOTE ]

As jason1990 pointed out in another thread, it's a bit bizzare for Sklansky to be giving this bogus argument against the Logical Probability school of thought because he seems to implicitely assume Logical Probability as part of many of his other arguments. At least he seems to lean in that direction. I suspect he doesn't really know what the foundation is for his probability reasoning other than supreme confidence in its Sklansky Rigor.

PairTheBoard

[borisp]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
There is a certain school of thought called logical probability which regards probabilities as extended truth values -- 0 is false, 1 is true, and anything in between is a "degree of plausibility."

[/ QUOTE ]
This "school of thought" is nonsense. However, to prove that it is nonsense, you cannot use nonsense.

[/ QUOTE ]
I may end up agreeing with you, but I am presently trying to remain open-minded. As far as I can tell, logical probability is alive and kicking, as far as philosophical interpretations of probability are concerned. I doubt that you, as someone who does mathematics and not philosophy, could unequivocally prove that an entire branch of philosophy is nonsense. However, if you would like to try, I might be interested to read what you have to write.

[/ QUOTE ]
First of all, I definitely grant that this definition of truth, as taking values in the interval [0,1], loosely behaving like probability, is of great practical use. If philosophers are able to achieve a common ground via this convention, then so be it. A mathematician would probably call it an "abuse of notation." I personally use this mode of reasoning with virtually every decision I make.

However, there simply does not exist a function that assigns to every potential description of every potential event a number that behaves according to the laws of probability. This is easily proven once one formulates a rigorous mathematical notion of probability. Look up nonmeasurable set on Wiki.

Essentially, the assumption that two rational people will draw the same conclusions with equal information is false, if you allow them to recursively make arbitrary choices. Your "DS playing a game" example actually demonstrates that. You could make the arbitrary choice to inform him of your prediction, but if you don't use this new knowledge of what you are about to do to calculate a new probability, then you are violating the methods you have set out to employ. In fact, there are an infinitude of arbitrary choices that you or any other rational being could employ to get "information from nothing." Succinctly put, logical probability assumes that every rational being's imagination will fail in exactly the same way, and this is incorrect.

(Alternatively, quantum physics offers us a real world verification that the future is not deterministic; i.e., choice #1 is the worst assumption in the three assumptions above? Any "multiple beam splitter" example should demonstrate this pretty clearly...try starting with the EPR paradox on Wiki, although I am pretty much guessing on this sort of thing.)

Again, the practicality of logical probability is not what I'm debating, but rather its rigorous mathematical footing. It is in this way that I use the word "nonsense," in that, mathematically speaking, it is nonsense. Once philosophers start discussing properties of a real valued function, they have moved into the realm of mathematics. Hence they must be held to the appropriate standards, or else admit that they are just using a convention that could collapse under rigorous scrutiny.

[borisp]

[ QUOTE ]
I suspect he doesn't really know what the foundation is for his probability reasoning other than supreme confidence in its Sklansky Rigor.

[/ QUOTE ]
This made me laugh...well put.

[David Sklansky]

[ QUOTE ]
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There is a certain school of thought called logical probability which regards probabilities as extended truth values -- 0 is false, 1 is true, and anything in between is a "degree of plausibility."

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This "school of thought" is nonsense. However, to prove that it is nonsense, you cannot use nonsense.

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As jason1990 pointed out in another thread, it's a bit bizzare for Sklansky to be giving this bogus argument against the Logical Probability school of thought because he seems to implicitely assume Logical Probability as part of many of his other arguments. At least he seems to lean in that direction. I suspect he doesn't really know what the foundation is for his probability reasoning other than supreme confidence in its Sklansky Rigor.

PairTheBoard

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My original post was a good example of my sloppiness regarding writing my intentions.

Jason was explaining some details of the difference between Bayesians and non Baysians. And putting me in the Baysian camp. Which I suspect I am. During this explanation he mentioned that some Baysians have been searching, unsuccessfully so far, for an algorithm that would incorporate all the information available and spit out a probability. Or something like that.

Even though I am probably a Baysian, when Jason said this I thought I detected the makings os a paradox. And I posted that suspicion. Jason replied that I should work on the idea.

When the specifics came to me I was far from sure it had no holes in it. But I wanted to throw it out there. But a more reaonable title would have been. "Jason-I think I might come up with that paradox argument I was talking about. Here it is." I'm sure Jason realized that although I don't believe he replied at the time.

Also the fact that I used myself as the one who was given the results of the supposed universal algorithm was neither here nor there. Boris is way off on that.

[borisp]

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Also the fact that I used myself as the one who was given the results of the supposed universal algorithm was neither here nor there. Boris is way off on that.

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File that under the "jerking your chain" category...I know the person in question might as well be arbitrary. I thought it was funny to imagine DS being so smart that he could trick a perfect machine.

I agree that your presentation is instructive towards the practical impossibility of such an algorithm.

But the logical fallacy I am pointing out here is widespread. "I can't think of why I am wrong" is not the same as "I am right." Put another way: "At this point, to complete my argument, I must appeal to a contradiction" is NOT a proof by contradiction. You are a high profile poster that can act as a vehicle for me pointing this out. You might understand the limitations of your argument and were presenting it in an intuitively clear fashion. But I bet that many readers confuse this for absolute truth.

[jason1990]

Proponents of logical probability do calculations in Kolmogorov's measure theoretic framework. There is nothing non-rigorous as far as their mathematics is concerned. In fact, even most subjectivists use standard probability spaces. The question is not about who is mathematically rigorous. They all are. The question is about philosophical interpretation.

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This is easily proven once one formulates a rigorous mathematical notion of probability...Essentially, the assumption that two rational people will draw the same conclusions with equal information is false

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I do not think there is an easy "proof" that this fundamental tenet of the logical probabilists is false. If you have one which involves simply some rigorous mathematics, then you are on your way to fame and fortune. (Well, at least fame.)

[jason1990]

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When the specifics came to me I was far from sure it had no holes in it. But I wanted to throw it out there. But a more reaonable title would have been. "Jason-I think I might come up with that paradox argument I was talking about. Here it is." I'm sure Jason realized that although I don't believe he replied at the time.

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I did realize that. I do not know if there is the seed of something in your OP or not. If there is, I am pretty sure it is not new. The idea that announcing a probability can change that probability is a simple one to grasp. It even appears in the science fiction work of Asimov. I do not know if and in what form that idea has been explored in the area of the philosophy of probability. Borodog mentioned Austrian economists. I do not know anything about Austrian economists, but perhaps there is a connection between that school of thought and logical probability. If so, that might be a place to look in order to see if something like this has been considered and what form it has taken.

[borisp]

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Proponents of logical probability do calculations in Kolmogorov's measure theoretic framework. There is nothing non-rigorous as far as their mathematics is concerned.

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The problem lies in the assumption that there exists a function, with domain the set of all event descriptions, and with range [0,1], such that the range values "behave like probability." There is no such function. This is what can be proven mathematically. The proof rests on the capability to make arbitrary choices, which is why I related it to your example in that manner.

If you deny that there is such a thing as an arbitrary choice, then that is a matter of philosophy, I agree, and then the mathematics is irrelevant. I would conjecture that the assumption that two independent rational beings always assign equal probabilities, given equal information, is equivalent to the assumption that a rational being is incapable of a wholly arbitrary choice. Mathematics can show the equivalence, but what is actually "true" about rational beings is the domain of the philosopher, and I concede this aspect.

Note that one is not prevented one from defining such a function, in a particular instance of a collection of event descriptions (i.e. with restricted domain), and performing perfectly rigorous math with it. This is what I think you mean when you say that they perform calculations in a rigorous framework.

It is my impression that philosophers do this sort of thing all the time...meaning they postulate that a certain object or method of comparison exists, satisfying certain properties, without demonstrating that it actually does. Axioms must be checked rigorously for consistency; the only way to do this is to actually demonstrate an item that satisfies your axioms.

[jason1990]

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The problem lies in the assumption that there exists a function, with domain the set of all event descriptions, and with range [0,1], such that the range values "behave like probability."

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In my reading so far, I have not seen anyone postulate the existence of such an object.

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I would conjecture that the assumption that two independent rational beings always assign equal probabilities, given equal information, is equivalent to the assumption that a rational being is incapable of a wholly arbitrary choice. Mathematics can show the equivalence

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I think you are overestimating the power and applicability of mathematics. If we had acceptable mathematical definitions of "independent rational beings," "wholly arbitrary choices," and so forth, then philosophers would be out of business.

[borisp]

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The problem lies in the assumption that there exists a function, with domain the set of all event descriptions, and with range [0,1], such that the range values "behave like probability."

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In my reading so far, I have not seen anyone postulate the existence of such an object.

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I tried to summarize your description of logical probability. Although "event descriptions" is probably a bad phrase, you can insert "propositions" if you like. It is the existence of this function, in principle, that the consistency of logical probability rests on.

Godel tells us that no consistent axiomatic framework, robust enough to encompass arithmetic, can prove its own consistency ("no news" is "good news"), so the absence of a proof of this object is not cause for despair. But you must recognize that consistency rests on its existence.

One common extension of the framework actually asserts its non existence (while obviously introducing new consistency problems, as you have "cleared up" the old ones). This extension is the Axiom of Choice, which when translated to logical probability, amounts to "whatever you define to be a rational being is capable of arbitrary choices."

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I would conjecture that the assumption that two independent rational beings always assign equal probabilities, given equal information, is equivalent to the assumption that a rational being is incapable of a wholly arbitrary choice. Mathematics can show the equivalence

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I think you are overestimating the power and applicability of mathematics. If we had acceptable mathematical definitions of "independent rational beings," "wholly arbitrary choices," and so forth, then philosophers would be out of business.

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You are probably overestimating what I claim...I am saying that there is an equivalence of assumptions regarding what rational means. This should only serve to highlight our lack of understanding of this notion, and it points to the real question "What, if anything, is rational?" A mathematician cannot answer this question, but he can tell when two seemingly different answers are the same.

Philosopher: Which came first, chicken or egg? Are such questions even meaningful?

Mathematician: Chicken if and only if egg. That's all I can say.

Philosopher: Ok, but that doesn't help. What are the rational consequences of perfect Bayesianism, logical probability, etc.?

Mathematician: The consistency of the practice of logical probability rests on the assumption that rational beings cannot make arbitrary recursive choices, so this must be accounted for if someone sets out to define rational. I have no idea how to do this, though.

If anything, you have pointed out to me what I am claiming when I claim that logical probability is nonsense; I am claiming that it is inconsistent with the Axiom of Choice.

I agree that the "truth" of the AoC is outside of the realm of mathematics. I suggest that logical probabilists discuss candidly its philosophical implications, as it is highly relevant to their discussion.

[jason1990]

I think you have gone down a road that is totally unrelated to the real issues in logical probability. It may be because of my inadequate explanation of the philosophy. I am but a beginning student of this philosophy. If you wish to criticize logical probability, you should probably read some of the logical probabilists. Here are some comments that you might find illuminating.

I think logical probabilists acknowledge the existence of ill-posed problems. That is, I think they accept that information is sometimes inadequate for producing a numerical probability. However, they still deny that information can produce two distinct numerical probabilities.

Logical probabilists (at least Jaynes, who I am currently reading) are not really interested in "perfectly rational beings." I used that because I thought it was the simplest way to give a loose description. If I had known it would be a source of confusion, I would have avoided it. Jaynes is interested in developing a body of reasoning principles, derived from a small set of what he calls "desiderata."

Really, though, if you want to find fault with this philosophy, then you should not take my posts as the official description of what it claims. Read Jaynes himself to start, if you want, and I will discuss it. I think parts of "Probability Theory: The Logic of Science" are available free online.

[FoxyJ]

David give it up already. You are a pig. I cant believe you would even mention brandi s name after the conversation you last had with her. After she got off the phone with you she called me crying. I feel somewhat bad about writing this. Im not really her friend but we ve hung out a few times and I think she is a really nice super cool girl. I heard about your whole
conversation about wanting to teach her the "way the poker world really is" and "how men really are" and how she'd be better off sleeping around and whoring herself out to freddy deeb or bobby baldwin so that she would never have to worry bout money and being staked etc. She called me crying and became very depressed. Thats why she has not taken your phone calls this past month or studied with you. I wont get into specific details with it here but for those of you who are eager to know he was basically pissed at her because she has been dating gus hansen since the world series this past summer. Im surprised no one mentioned anything about it on NVG. Every poker regular at Bellagio esp bobbys room knew. gus was not at all secret in his affection for her and they were seen dining together and making out in hotel elevators all summer long. From what I understand it was the best sex of her entire life and they respect and care for each other although they have a mutual understanding and open relationship. And they even have a video of one of their escapades.

[mrick]

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Logical probabilists (at least Jaynes, who I am currently reading) are not really interested in "perfectly rational beings."

[/ QUOTE ]Economists are.




You probably dont speak much Austrian. Pity.