Derivatives of Entropy Rate in Special Families of Hidden Markov Chains

TL;DR

This paper extends the understanding of entropy rate derivatives in hidden Markov chains, especially for a class called 'Black Holes,' providing explicit formulas and stabilization results for derivatives at extreme noise levels.

Contribution

It generalizes previous results to a new class of hidden Markov chains and offers an explicit formula for the first derivative of the entropy rate in these models.

Findings

Derivatives of entropy rate stabilize at finite times for Black Hole chains.

Explicit formulas for the first derivative of entropy rate in binary symmetric noise cases.

Generalization of previous results to broader classes of hidden Markov chains.

Abstract

Consider a hidden Markov chain obtained as the observation process of an ordinary Markov chain corrupted by noise. Zuk, et. al. [13], [14] showed how, in principle, one can explicitly compute the derivatives of the entropy rate of at extreme values of the noise. Namely, they showed that the derivatives of standard upper approximations to the entropy rate actually stabilize at an explicit finite time. We generalize this result to a natural class of hidden Markov chains called ``Black Holes.'' We also discuss in depth special cases of binary Markov chains observed in binary symmetric noise, and give an abstract formula for the first derivative in terms of a measure on the simplex due to Blackwell.