A cautious user's guide in applying HMMs to physical systems

Nature, as far as we know, evolves continuously through space and time. Yet, the ubiquitous hidden Markov model (HMM)-originally developed for discrete time and space analysis in natural language processing-remains a central tool in interpreting time series data drawn from physical systems. This raises a fundamental question: What are the implications of applying a discrete-state, discrete-time framework to analyze data generated by a continuously evolving system? Through synthetic data generated using Langevin dynamics in an effective potential, we explore under what circumstances HMMs yield interpretable results. Our analysis reveals that the discrete-state approximation acts primarily as an abstraction. As such, the inferred states visited in time often more closely reflecting the measurement protocol and modeling choices than features of the underlying physical potential. Crucially, we demonstrate that the states visited over the course of a time series recovered by the HMM can be tuned a priori by adjusting the data acquisition scheme. In fact, it is even possible to misleadingly and reproducibly recover "intermediate" states using different HMM tools for a system evolving in a single well potential. We conclude with a note of measured caution: while HMMs offer a mathematically elegant framework for time series inference, their use in physical modeling should be guided by an awareness of their limitations. In this light, we outline important generalizations of the HMM to continuous space and time and highlight the importance of a well calibrated measurement noise model.