Analyticity of Entropy Rate of Continuous-State Hidden Markov Chains

TL;DR

This paper proves that the entropy rate of continuous-state hidden Markov chains is jointly analytic in the parameters of the input Markov chain and the channel, under mild assumptions, with applications to Cauchy and Gaussian channels.

Contribution

It establishes the joint analyticity of the entropy rate for a broad class of hidden Markov models, including specific channels like Cauchy and Gaussian, under mild conditions.

Findings

Entropy rate is jointly analytic in model parameters.

Analyticity holds for Cauchy and Gaussian channels.

Provides theoretical foundation for analyzing hidden Markov chains.

Abstract

We prove that under certain mild assumptions, the entropy rate of a hidden Markov chain, observed when passing a finite-state stationary Markov chain through a discrete-time continuous-output channel, is jointly analytic as a function of the input Markov chain parameters and the channel parameters. In particular, as consequences of the main theorems, we obtain analyticity for the entropy rate associated with representative channels: Cauchy and Gaussian.