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.

Inferring models from samples of stochastic processes is challenging, even in the most basic setting in which processes are stationary and ergodic. A principle reason for this, discovered by Blackwell in 1957, is that finite-state generators produce realizations with arbitrarily-long dependencies that require an uncountable infinity of probabilistic features be tracked for optimal prediction. This, in turn, means predictive models are generically infinite-state. Specifically, hidden Markov chains, even if finite, generate stochastic processes that are irreducibly complicated. The consequences are dramatic. For one, no finite expression for their Shannon entropy rate exists. Said simply, one cannot make general statements about how random they are and finite models incur an irreducible excess degree of unpredictability. This was the state of affairs until a recently-introduced method showed how to accurately calculate their entropy rate and, constructively, to determine the minimal set of infinite predictive features. Leveraging this, here we address the complementary challenge of determining how structured hidden Markov processes are by calculating the rate of statistical complexity divergence-the information dimension of the minimal set of predictive features.

Maxwellian ratchets are autonomous, finite-state thermodynamic engines that implement input-output informational transformations. Previous studies of these "demons" focused on how they exploit environmental resources to generate work: They randomize ordered inputs, leveraging increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir while respecting both Liouvillian state-space dynamics and the second law. However, to date, correctly determining such functional thermodynamic operating regimes was restricted to a very few engines for which correlations among their information-bearing degrees of freedom could be calculated exactly and in closed form, a highly restricted set. Additionally, a key second dimension of ratchet behavior was largely ignored: ratchets do not merely change the randomness of environmental inputs, their operation constructs and deconstructs patterns. To address both dimensions, we adapt recent results from dynamical-systems and ergodic theories that efficiently and accurately calculate the entropy rates and the rate of statistical complexity divergence of general hidden Markov processes. In concert with the information processing second law, these methods accurately determine thermodynamic operating regimes for finite-state Maxwellian demons with arbitrary numbers of states and transitions. In addition, they facilitate analyzing structure versus randomness tradeoffs that a given engine makes. The result is a greatly enhanced perspective on the information processing capabilities of information engines. As an application, we give a thorough-going analysis of the Mandal-Jarzynski ratchet, demonstrating that it has an uncountably infinite effective state space.

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Extended acoustic interactions with a humpback whale (Megaptera Novaeangliae) were captured via playbacks of the purported "whup/throp" social call and hydrophone recordings of the animal's vocalized responses during August 2021 in Frederick Sound, Southeast Alaska. Fluke photographs identified the animal as a female named Twain (HappyWhale.com identity SEAK-0401) first observed some 35 years ago. We document Twain's life history via sightings made over three decades, reporting as much as is known (and allowed for public distribution) about Twain. The observational history gives illuminating snapshots of the long history of the individual behind the acoustic interactions.