March 3, 2026

Advanced Estimation of Hidden Markov Models: Handling Discrete Latent Variables in Prediction and Forecasting

Key Points

Prediction and forecasting improve with hidden Markov models that utilize advanced latent variable strategies, addressing challenges like missing values.
The expectation-maximization algorithm is standard but may struggle with local maxima, necessitating alternative optimization methods.
Incorporating a tempered EM algorithm and an evolutionary algorithm can enhance model convergence towards the global maximum.
Variable selection not only boosts interpretability but also enhances predictive accuracy, which is crucial for longitudinal data modeling.

Abstract

Hidden Markov (HM) models are widely used for modeling longitudinal data. Their estimation remains challenging due to missing values, multimodal likelihood functions, and the need for variable selection. In this work, we present both basic and advanced formulations of HM models with discrete latent variables, focusing on maximum likelihood estimation. The expectation–maximization (EM) algorithm is introduced as the standard approach, and its limitations in the presence of local maxima are highlighted. We further consider a tempered EM algorithm, as well as an evolutionary algorithm that enables a more comprehensive exploration of the parameter space and improves convergence toward the global maximum. Special attention is devoted to handling partially missing observations under the missing-at-random assumption and to incorporating variable selection strategies that enhance both interpretability and predictive accuracy. Finally, we discuss the role of HM models in prediction and forecasting, showing both short-term and long-term predictive performance. The models are illustrated with both simulated and real data applications.

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