Entropy-Rate Selection for Partially Observed Processes

TL;DR

This paper develops a theoretical framework for maximizing entropy rate in partially observed stochastic processes, establishing existence, uniqueness, and structural properties of the maximizer under various constraints.

Contribution

It introduces a novel entropy-rate maximization approach for observable processes, providing structural results, optimality conditions, and a latent variable interpretation.

Findings

Proves existence and uniqueness of the entropy-rate maximizer.

Characterizes the maximizer under fixed marginal and fixed r-block law constraints.

Derives optimality conditions and local geometric properties of the maximizer.

Abstract

I formulate an entropy-rate maximization problem at the observable level for stochastic processes observed through an information-reducing observation map. For a visible stationary law, the map determines an observational fiber of hidden stationary laws generating that law. In the finite-state finite-memory setting, retained visible constraints determine a feasible class of stationary $(r+1)$-block laws, and the entropy maximizer is defined as the entropy-rate maximizer on this class. The paper formulates entropy-rate maximization on feasible classes induced by partial observability and develops a structural theory for the resulting maximizer. I prove existence and uniqueness of the maximizer, with uniqueness under a fixed-context-marginal hypothesis and, more generally, via a strict-concavity characterization by row proportionality. Two global characterization regimes are central: a fixed one-point marginal yields the i.i.d. maximizer, and a fixed $r$-block law yields the $(r-1)$-step Markov extension. The gap functional equals a conditional mutual information and vanishes exactly at the maximizing completion. I also derive optimality conditions, local geometry of the maximizer, a latent random-mapping realization that leaves the visible law unchanged, and a local empirical consistency theorem, and illustrate the framework by an aliased hidden-state example.