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I agree that the envelope paradox uses standard mathematical tools to arrive at two opposite conclusions. Mathematics must be consistent, or you can prove everything, so we have to rule out one or the other chain of logic. In order to do that, we have to rule out calculations that are used every day in statistics ... My beef is with people who insist mathematics dictate the choice of assumptions. That's wrong. There is no reason outside of personal preference to prefer one approach over the other.
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I still don't understand what these two "approaches" are. Here's one approach: we assume that
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[1] if A and B are events, then P(A | B) = P(A and B)/P(B),
[2] if A and B are independent events, then P(A | B) = P(A),
[3] if A and B are mutually exclusive events, then P(A or B) = P(A) + P(B),
[4] if A is an event, then 0 <= P(A) <= 1.[/list]As far as I can tell, if we assume these four things, then it cannot be the case that
P(Y = 2X | X = k) = 0.5
for all k.
What is the other approach? Which of these assumptions does it reject? Does it involve one of these other "alternative" probability theories you mentioned? Because as far as I can tell, the work of Shannon and Arrow-Debreu was within the context of Kolmogorov's formalism, and Savage's main departure from the formalism was his use of "measures" which were only finitely additive. But the above assumptions do not require countable additivity, nor do they even require sigma-algebras, only algebras. You also suggest that this other approach, whatever it is, is used every day in statistics. So are you saying that there are mainstream statisticians who reject one or more of the above assumptions on a daily basis?
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The above quote is from my last post. After I posted it, I thought a little more and I think you already told us what the other approach is. You said,
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I'd rather do the math the natural way and trust that someone will figure out how to make it rigorous someday. Applied mathematical practice has always been ahead of theory, and is usually (but not always) justified in the end.
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So the first approach is the one I outlined above which tells us that the conditional probability of having the larger envelope, given that we observe k in the chosen envelope, cannot be 0.5 for all k.
The second approach is to have no formalism at all. To just do what feels right, and worry about formal justification later. This is the approach that tells us we should always switch, regardless of what is in the envelope.
In the second approach, not only can we not justify the steps, but they contradict assumptions [1]-[4]. And we are led to a conclusion which is intuitively absurd. But the method is so natural, it must be right, and someone will probably be able to rigorously justify it in the future. Moreover, there is no reason outside of personal preference to prefer one approach over the other.
If I met a statistician who thought this way, I wouldn't trust him with my piggy bank. I really hope that I'm wrong and that the "other approach" has something to do with Von Neumann.