Dealing with separation problem in hidden Markov models with covariates based on a penalized maximum likelihood approach

A penalized maximum likelihood estimation approach is proposed for discrete-time hidden Markov models where covariates affect the observed responses and serial dependence is considered. The proposed penalized maximum likelihood method addresses the issue of latent state separation that typically occurs when this model is applied to binary and categorical response variables with a limited number of categories, resulting in extremely large estimates of the support points of the latent variable assumed with a discrete, left unspecified distribution. We also propose a cross-validation approach for jointly selecting the number of hidden states and the roughness of the penalty term. The proposal is illustrated through a simulation study comparing parameter estimation accuracy and computational efficiency across different estimation procedures. We also demonstrate the potential of this class of models through the analysis of longitudinal data collected during spinal anesthesia to monitor the occurrence of hypotension in patients, and we compare the results with those obtained from other standard models.