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

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.