Analyticity of Entropy Rates of Continuous-State Hidden Markov Models

TL;DR

This paper proves that the entropy rates of continuous-state hidden Markov models are analytic functions of model parameters, under mild conditions, aiding inference and system identification.

Contribution

It establishes the analyticity of entropy and relative entropy rates for a broad class of continuous-state hidden Markov models using the analytic continuation principle.

Findings

Entropy rates are analytic functions of parameters.

Results apply to many practical hidden Markov models.

Supports advanced statistical inference and system identification.

Abstract

The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is shown for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several classes of hidden Markov models met in practice. These results are relevant for several (theoretically and practically) important problems arising in statistical inference, system identification and information theory.