Necessary and sufficient conditions for reliability of posterior matching in arbitrary dimensions

The posterior matching (PM) scheme is a mutual information maximizing scheme for efficiently communicating a message point in a continuum over a noisy channel with feedback. It was recently developed when the message point was on a subset of the real line, and we more recently have generalized the framework to arbitrary dimensions with the theory of optimal transport. Although the PM scheme is guaranteed to maximize mutual information, in some situations it is not reliable (e.g. the posterior distribution does not converge to a point mass). Here, we show that the PM scheme is reliable if and only if the nonlinear filter of a certain hidden Markov model achieves maximal accuracy. We show that ergodicity of a random process pertaining to the channel input process is necessary for PM reliability. Lastly, we show that joint ergodicity of random processes pertaining to inputs and outputs of the channel is a sufficient condition for reliability. The latter results leverages optimal transport properties (invertibility) of the encoder map.