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.

隐藏的马尔可夫链是广泛应用的随机过程统计模型，从基础物理和化学到金融，健康和人工智能。但是，即使链是有限状态的，它们所产生的隐马尔可夫过程也非常复杂：由于其预测特征的集合通常是无限的，因此不存在有关香农熵率的有限表达式。因此，迄今为止，人们无法就它们的随机性和结构性做出一般性的陈述。在这里，我们通过展示如何有效和准确地计算其熵率来解决这一挑战的第一部分。我们还将展示该方法如何提供最小的无限预测特征集。续集解决了挑战的第二部分。