  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.
