Nearest symmetric distributions

Given a finite set S and a group G acting on S, a probability distribution on S is G-invariant, or symmetric, if it is constant on the orbits of the group action. How can one symmetrize a non-symmetric probability distribution - i.e. find the “nearest” G-invariant distribution? When would one want to? We find that for the Rényi and Bregman divergences, symmetrization means averaging over the group orbits in a natural way; for the special case of the Kullback-Leibler divergence, relevant in maximum likelihood inference and in large deviations, symmetrization is either the arithmetic or geometric average, depending on the order of arguments. We apply our results for the symmetries of time-reversibility and exchangeability for Markov chains to answer questions in inference and in large deviations: given some data, what is the maximum likelihood time-reversible Markov chain? How long must we “watch” a trajectory from a Markov chain to establish whether time is running forward or in reverse? What does the data look like conditioned on a time-reversal fluctuation?