Entropy rate calculations of algebraic measures

TL;DR

This paper derives exact formulas for the entropy rate of certain hidden Markov processes using algebraic measures, providing bounds on approximation errors and numerical computations for small models.

Contribution

It introduces a method to compute the entropy rate of hidden Markov processes via algebraic measures under irreducibility conditions, with practical error bounds and numerical results.

Findings

Exact formulas for entropy rates of hidden Markov models.

Upper bounds on approximation errors.

Numerical entropy calculations for small models.

Abstract

Let $K = \{0,1,...,q-1\}$. We use a special class of translation invariant measures on $K^\mathbb{Z}$ called algebraic measures to study the entropy rate of a hidden Markov processes. Under some irreducibility assumptions of the Markov transition matrix we derive exact formulas for the entropy rate of a general $q$ state hidden Markov process derived from a Markov source corrupted by a specific noise model. We obtain upper bounds on the error when using an approximation to the formulas and numerically compute the entropy rates of two and three state hidden Markov models.