Information Structures for Feedback Capacity of Channels With Memory and Transmission Cost: Stochastic Optimal Control and Variational Equalities

Abstract

Stochastic optimal control theory and a variational equality of directed information are applied, to develop a methodology to identify the information structures of optimal channel input conditional distributions, which maximize directed information, for classes of channel conditional distributions and transmission cost functions that depend on previous channel output symbols. The subsets of the maximizing distributions are characterized by conditional independence. One of the main theorems of this paper states that, for any channel conditional distribution with finite memory on past channel outputs, subject to an average cost constraint, then the information structure of the optimal channel input conditional distribution, which maximizes directed information, is determined by the maximum of the memory of the channel distribution and the functional dependence of the transmission cost function on past channel outputs. This theorem provides, for the first time, a direct analogy, in terms of the conditional independence properties of maximizing distributions, between the characterization of feedback capacity of channels with memory, and Shannon's two-letter characterization of capacity of memoryless channels. Another main result of the paper is the identification of sufficient conditions for the validity of direct and converse coding theorems, for unstable Gaussian channel models with memory, that is based on the ergodic theory of Markov decision.