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Has the idea of a maximum entropy probability distribution been explored for function spaces, and if so what are some key papers, books, or terms to look for?

For $\mathbb{R}^n$ (and discrete spaces), the problem appears to be well studied - one maximizes the quantity, $$-\int f(x) ~ log f(x) ~ dx,$$ over the set of feasible candidate densities $f$, where the integral is taken with respect to the standard Lebesgue measure in $\mathbb{R}^n$ or counting measure in discrete spaces. There seems to be much literature about this problem which goes under names such as "non-informative priors", "maximum entropy distributions", "Jeffrey's priors", and the like.

However, I've found little on this topic in the infinite dimensional (function space) setting. Can concept of maximum entropy priors be generalized to function spaces, or is the idea of entropy fundamentally incompatible with infinite dimensional space?

Note: this thread wasn't getting answers here so I reposted at mathoverflow.

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  • $\begingroup$ There is now two answers to the mathoverflow version! $\endgroup$ Aug 15 '18 at 16:09

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