# Maximum entropy priors in infinite dimensional spaces

Is there an extension of maximum entropy probability distributions for function spaces?

For $\mathbb{R}^n$ and discrete spaces, there is much literature about this problem under names such as "non-informative priors", "maximum entropy distributions", "Jeffrey's priors", and the like.

There is an extension to locally compact topological groups, where the Haar measure $U$ takes the place of the Lebesgue measure, and one looks for measures $P$ minimizing the information divergence, $$D(P||U):= \begin{cases} \int log \frac{dP}{dU} dP, & \text{ if } P\ll U; \\\ \infty, & \text{else.}\end{cases}$$

However, I've found little about this in the infinite dimensional setting. Can the concept of maximum entropy priors be generalized to (some class of) function spaces, or is the idea of entropy fundamentally incompatible with spaces that are not locally compact?

Notes,

• Thanks, this was very helpful. I've looked around a lot over the last couple weeks and my conclusion is that exponential families are the most natural generalization of maximum entropy distributions to infinite dimensional spaces. To future readers, it is instructive to consider the finite dimensional case with absolutely continuous distributions, where (under mild conditions) one can explicitly show that exponential families extremize the constrained convex optimization problem $max_f H(f)$ such that $\mathbb{E}_f(g_i)=t_i$. Nov 1 '12 at 5:56