Shannon Entropy Rate of Hidden Markov Processes

TL;DR

This paper presents an efficient method to calculate the Shannon entropy rate of hidden Markov processes, addressing a fundamental challenge in understanding their randomness and structure.

Contribution

It introduces a novel approach to accurately compute entropy rates and identify the minimal set of predictive features for hidden Markov processes.

Findings

Efficient calculation of entropy rates for hidden Markov processes

Identification of the minimal set of infinite predictive features

Advancement in understanding the structure of complex stochastic processes

Abstract

Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their Shannon entropy rate exists, as the set of their predictive features is generically infinite. As such, to date one cannot make general statements about how random they are nor how structured. Here, we address the first part of this challenge by showing how to efficiently and accurately calculate their entropy rates. We also show how this method gives the minimal set of infinite predictive features. A sequel addresses the challenge's second part on structure.