The envelope problem, and a possible solution

[PairTheBoard]

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Practicing Bayesian statisticians use the same rigorous mathematics (in particular, the same definition of a probability space) as we probabilists do. Here's a toy example inspired by poker.

Let m be my winrate in BB/100 and s my standard deviation. I could model the outcome of a single 100 hand session by

X = m + sN,

where N is a Normal(0,1). Suppose I know that s=15, but I don't know m. However, I have "beliefs" about m. I believe that m=2. Of course, I'm not certain of it, it could be higher or lower. But I am fairly certain that I am at least a winning player. I might decide to model these beliefs by saying that

m = 2 + M,

where M is a Normal(0,1), independent of N. In "reality", m is a fixed, deterministic number, but I am representing it as a random variable in order to model my beliefs. Similarly, in the envelope paradox, the amounts in the envelopes are fixed, deterministic values, but we may choose to represent them as random variables in order to model our beliefs about the contents of those envelopes.

Now suppose I play a session of poker and lose 100 BB. I now need to adjust my "beliefs". This, of course, is analogous to the act of opening one of those envelopes. The adjustment is done by computing the conditional law of m, given X, and plugging in the observed value X=-100. We get the conditional CDF by computing

P(m <= x | X).

Upon computing this expression and differentiating, we find that the conditional law of m is again normal. According to my hasty calculations, its mean is

(225/226)*2 + (1/226)*X

and its variance is simply 225/226 (regardless of the value of X). Plugging in X=-100, our updated "belief" about our winrate is that it is 350/226=1.55 and our "belief" has a standard deviation of about 0.998.

Obviously, the process gets more complicated when we try to apply it to a sequence of sessions, and we might find it useful to apply the mathematical tools of filtering theory. One aspect of this that I find interesting, as it relates to poker, is that you can get analogs of confidence intervals which appear much more realistic over a smaller sample size. Of course, those estimates always contain some residue of our original "beliefs". For what it's worth, that residue goes away in the limit as the sample size increases.

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Great explanation. Thanks Jason. That's something I can understand. So, is there some controversy about that technique? What is the nature of the Bayesian justification for the technique?


PairTheBoard