A few weeks ago I read this. David Blei and I spent some time thinking hard about this a few years back (thanks to Kary Myers for pointing us to it):
In short I was thinking that Ã¢â‚¬Å“bayesian belief updatingÃ¢â‚¬Â and Ã¢â‚¬Å“maximum entropyÃ¢â‚¬Â were two othogonal principles. But it appear that they are not, and that they can even be in conflict !
Example (from Kass 1996); consider a Die (6 sides), consider prior knowledge E[X]=3.5.
Maximum entropy leads to P(X)= (1/6, 1/6, 1/6, 1/6, 1/6, 1/6).
Now consider a new piece of evidence A=Ã¢â‚¬ÂX is an odd numberÃ¢â‚¬Â
Bayesian posterior P(X|A)= P(A|X) P(X) = (1/3, 0, 1/3, 0, 1/3, 0).
But MaxEnt with the constraints E[X]=3.5 and E[Indicator function of A]=1 leads to (.22, 0, .32, 0, .47, 0) !! (note that E[Indicator function of A]=P(A))
Indeed, for MaxEnt, because there is no more Ã¢â‚¬Ëœ6Ã¢â‚¬Â², big numbers must be more probable to ensure an average of 3.5. For bayesian updating, P(X|A) doesnÃ¢â‚¬â„¢t have to have a 3.5 expectation. P(X) and P(X|a) are different distributions.
Conclusion ? MaxEnt and bayesian updating are two different principle leading to different belief distributions. Am I right ?
I don’t believe there is any paradox at all between MaxEnt (perhaps more generally, MinRelEnt) and Bayesian updates. Here, straight MaxEnt make no sense. The implication of the problem is that the ensemble average 3.5 is no longer an active constraint. That is, we no longer believe the contraint E[X]=3.5 once we have the additional data that X is an odd number. The sequential update using minimum relative entropy is identical to Bayes rule and produces the correct answer. These two answers are simply (correct) answers to different questions.