The envelope problem, and a possible solution

[AaronBrown]

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I can see that you want to use the Paradox as a vehicle for discussing the (important) philosophical issues at the heart of statistical inference. In my opinion, this is an excellent use of the Paradox. But rather than coming right out and saying that is what you are doing, you phrase things in a way that can lead the less informed to believe that the branch of mathematics which we call probability theory is logically inconsistent.

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Oh, sorry, thank you for putting it so clearly. I should have said this at the beginning. I believe the paradox was invented and discussed by people interested in the application of probability theory to real problems. No one disputes that you can get a consistent theory of probability, the open issue is whether that consistent theory is rich enough to answer useful questions.

In fact, we have not one, but four consistent theories of probability, due to Shannon, Arrow-Debreu, Savage and Von Neumann. Savage was a great Bayesian, Shannon was the most objectivist (his theory of probability is popular in quantuum mechanics, possibly the only true physical randomness). Shannon's the guy who built a mechanical hand to flip coins reliably heads or tails (Ed Thorp saw it and immediately made the connection to roulette, the two of them built wearable computers to predict spins in a casino). The other two proposed abstract probabilities that were objective in principle but required conceptual experiments to define. No doubt there are other consistent theories as well, that never attacted major followings.

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. Statistics can survive these radical surgeries, but I think they are foolish. Bayesians reject useful tools, Frequentists engage in tedious hair-splitting. 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.

The batting average question is also pure mathematics. The question isn't about real baseball players. It asks: What is the probability that two automotons with constant, independent and equal probabilities of getting hits would differ in batting average as much as the two players in question? The answer is the same if the automotons flip coins or spin roulette wheels, only the probabilities matter. The use of this answer in inference, of course, does involve baseball.

I also agree that the paradox does not specify mathematical reasoning at all, and that is key. It asks a common sense question that appears to have two opposing common sense answers. People layer the mathematics on to it. That's fine, but making technical assumptions to make one argument right and one wrong doesn't answer the fundamental question of the paradox.

My beef is not with people who make one set of assumptions or another, although I prefer to live with inconsistency. 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.

Consider the analogy to Godel's famous result of that mathematics cannot be both consistent and complete. He makes some assumptions, such as that mathematics includes counting numbers. Some people have tried to rescue a complete consistent mathematics by eliminating counting numbers. That might be a useful intellectual exercise (probably not) but it clearly takes mathematics far away from what people want to use it for. Most mathematicians accepted the result and decided to live with an incomplete mathematics, in which there were true, false and undecidable statements. The universe survived.