Shannon Entropy Rate of Hidden Markov Processes

Alexandra M. Jurgens and James P. Crutchfield

Complexity Sciences Center
Physics Department
University of California at Davis
Davis, CA 95616

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.

Alexandra M. Jurgens and James P. Crutchfield, “Shannon Entropy Rate of Hidden Markov Processes” Journal of Statistical Physics 183:32 (2020) 1-18.
[pdf]. [cond-mat.stat-mech].