[PairTheBoard]

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You seem to treat this as a paradox designed by people who have envelopes and don't know whether or not to switch.

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Not at all.

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It is in fact a paradox designed to demonstrate that common statistical reasoning can lead to contradictory conclusions.


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I would say it's designed to show the need to be careful when applying mathematical tools like expected value and probabilties. When you speak in those terms you need to have a clear mathematical model in mind to which you are applying them.

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Bayesian statistics defines probability as subjective belief.

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This is why Mathematical Probabilists and Statisticians don't attend conferences of such so called Bayesians. I would define probability as a branch of mathematics and expect the full rigor of mathematics in its application. There's nothing subjective in such rigor.

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To a Bayesian, showing that no consistent prior distribution justifies switching regardless of the amount you observe, shows that always switching is irrational. My objection is that in real decisions, people often don't have consistent prior distributions.


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I really don't know much about the Bayesians. My impression is that they sort of create a prior distribution out of thin air, do sampling, then apply actual mathematical Bayesian techniques to adjust the original distribution depending on the sample. They then claim to have evidence for the real original distribution when what they really have are conclusions based on their original assumptions.

If that's the case I don't blame you for having a bone to pick with them. However I don't think it's central to the two Envelope Paradox. In the 2E paradox people are simply not being careful in how they apply the mathematical tools of probabilty and expectation.

However, if you'd like to explain more fully and clearly what Bayesians do and give examples of real life situations where you would disagree with their techniques due to "no consistent prior distribution" - whatever that means - I'd be interested in reading it.

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It's not bad math to compute expected value in different units from someone else. Everyone has different units if you look carefully enough.


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You misquoted me here. It was "variable" units I objected to. Computing expected value with a unit of measure that changes over time and treating it as if it's fixed. That's not only bad math but presenting conclusions from such a calculation to make your case is downright misleading.

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"PTB -
There's no need to assume (b). (b) is simply a true statement. Also, if by (a) you mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then that is simply a true statement as well."

You insist both statements are true and regard them as not only beyond argument, they are so obvious, they are beyond the need to state as assumptions. The paradox was designed to teach you they cannot be both true all the time.


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Not at all. The paradox was designed to force you to make the statements in a clear way and to understand what you mean by them. When you do so they are both true as I showed in my last post. The paradox forces you to see where you mistakenly use them to jump to a false conclusion as I also showed in that post. If you are going to quote me, quote this rather than the phrase above out of context.

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PTB -
" if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then there is a repeatable experiment you have in mind that can prove out this statement of probabilty. The experiment is for numerous people to pick one of the two envelopes at random. The more people who pick, the closer you will see half picking the higher amount and half the lower amount. There's a 50-50 chance that any one of them will pick the higher amount."


This is the frequentist argument. If that's your definition of probability, then (b) is not always true. This is the criticism that led to Bayesians to reject that definition of probability. Bayesians achieve consistency by rejecting (a) instead.


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I am not defining probabilty by way of frequencies. Probabilty has rigorous mathematical definitions. However, out of those definitions comes the law of large numbers which produces such frequencies. This is mathematical probabilty which is what we are usually talking about in this forum. If the Bayesians work with something else they call subjective probability then you need to clearly explain what they do and why it matters.

You say, "This is the frequentist argument. If that's your definition of probability, then (b) is not always true."

Yet look at (b) as you stated it,

(b) if you have a 50/50 chance to win $2 or lose $1 you have positive expected EV.

I challenge you to show this as not true under a standard mathematical definition of probabilty - where the law of large numbers - aka frequency - holds.

You say, "This is the criticism that led to Bayesians to reject that definition of probability. Bayesians achieve consistency by rejecting (a) instead."

Then what is a Bayesian definition of a probabilty space? I can give you a standard mathematical definition of a probabilty space for which the law of large numbers - aka frequencies - will hold. Having rejected this, What is the Bayesian definition of a probabilty space? Do they actually have something they can do math on?


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I understand the desire to make both (a) and (b) always true, but no one has discovered a way to do it.


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I think I have done it simply by stating them clearly and understanding what they mean. I agree though that there's no way that all misinterpretations of them can be made to be true.

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When you have nothing in one of the envelopes, (b) no longer applies. It only covers cases of $1 and $2, not $1 and $0. You can't reconstruct the paradox in this case.

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You missed my point here. If the Bayesians reject (a) on principle then they should reject it in this case as well. Yet it's clear there's no good reason to reject a clearly stated and well understood (a) that says all participants choosing one of the two envelopes at random have a 50-50 chance of choosing the $1 envelope. Just because Mr Lucky opens an envelope, sees it's empty, and now has a better than 50% chance of getting the $1 envelope if he switches there's no good reason to reject the statement (a) that future participants have a 50-50 chance of choosing a $1 envelope. Yet the Basesians should do so according to you on the same principle they rejected it in the 2E case, because (a) can't always be true? Because (a) is no longer true for Mr Lucky? Right. Because (a) can no longer be true when misconstrued.


PairTheBoard