Entropy of Hidden Markov Processes via Cycle Expansion

TL;DR

This paper introduces a novel method using cycle expansion of the zeta-function from dynamical systems theory to exactly compute the entropy and large deviation probabilities of certain Hidden Markov Processes.

Contribution

It applies cycle expansion techniques to derive exact formulas for entropy and moment-generating functions of HMPs, advancing understanding of their probabilistic properties.

Findings

Exact entropy calculations for a class of HMPs

Estimation of large deviation probabilities using Chernoff bound

Representation of entropy and moment-generating functions as convergent series

Abstract

Hidden Markov Processes (HMP) is one of the basic tools of the modern probabilistic modeling. The characterization of their entropy remains however an open problem. Here the entropy of HMP is calculated via the cycle expansion of the zeta-function, a method adopted from the theory of dynamical systems. For a class of HMP this method produces exact results both for the entropy and the moment-generating function. The latter allows to estimate, via the Chernoff bound, the probabilities of large deviations for the HMP. More generally, the method offers a representation of the moment-generating function and of the entropy via convergent series.