# When memory pays: Discord in hidden Markov models

## TL;DR

This paper investigates when maintaining a memory of past observations in hidden Markov models becomes beneficial by analyzing phase transitions and critical points through analytical methods and mappings to Ising models.

## Contribution

It introduces a discord order parameter to distinguish state estimates and analytically determines the critical point where memory improves inference in hidden Markov models.

## Key findings

- Identifies phase transitions in hidden Markov models with varying states and symbols.
- Derives the critical point analytically where memory becomes advantageous.
- Maps hidden Markov models to Ising models for deeper insight into phase transitions.

## Abstract

When is keeping a memory of observations worthwhile? We use hidden Markov models to look at phase transitions that emerge when comparing state estimates in systems with discrete states and noisy observations. We infer the underlying state of the hidden Markov models from the observations in two ways: through naive observations, which take into account only the current observation, and through Bayesian filtering, which takes the history of observations into account. Defining a discord order parameter to distinguish between the different state estimates, we explore hidden Markov models with various numbers of states and symbols and varying transition-matrix symmetry. All behave similarly. We calculate analytically the critical point where keeping a memory of observations starts to pay off. A mapping between hidden Markov models and Ising models gives added insight into the associated phase transitions.