# Why is Beta the maximum entropy distribution over Bernoulli's parameter?

Why is Beta(1,1) the maximum entropy distribution over the bias of a coin expressed as a probability given that:

• If we express the bias as odds (which is over the support $[0, \infty)$), then Beta-prime(1,1) is the corresponding distribution to Beta(1,1). Isn't the maximum entropy distribution over the positive reals the exponential distribution (which is not Beta-prime(1,1))?

• If we express the bias in log odds (which is over the support of the reals), then the logistic distribution (with mean 0 and scale 1) is the corresponding distribution to Beta(1,1).

Beta(1,1) makes sense as maximum entropy because it's flat over its support. The other distributions are not flat. If we had chosen a different parametrization, we should clearly arrive at the corresponding distribution (not something else). How are the other two distributions the maximum entropy distributions over their support? There must be some other requirement that I'm missing. What is it?

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You might want to cite some sources for the confusion. My guess is the underlying measure is not uniform over the real interval you are working with. –  John Jiang Aug 29 '10 at 3:46
The exponential distribution is the one that maximizes entropy SUBJECT TO a constraint on the expected value. –  Michael Hardy Aug 29 '10 at 5:00
Right, thanks. So, what's the constraint in this case? Surely, our expected odds is 1, but Beta-prime(1,1) is not the same distribution as Exponential(1). –  Neil Aug 29 '10 at 18:44
John, what do you mean cite sources for the confusion? –  Neil Aug 29 '10 at 18:46