Maximum likelihood inference for hidden Markov models with parsimonious parametrizations of transition matrices

In longitudinal data analysis, hidden Markov (HM) models are fundamental tools, especially when the analysis is focused on transitions or the need to cluster individuals dynamically. When individual covariates are available in the dataset, a typical problem is how to parametrize the transition probabilities based on these covariates in a parsimonious way. In fact, standard multinomial parametrizations of these probabilities lead to models with many parameters, which are also difficult to interpret and, consequently, to unstable parameter estimates. To overcome the above problems, different parametrizations of the transition probabilities of HM models with covariates are introduced based on multinomial logit models formulated by two different choices of the reference state of each logit. These parametrizations rely on constraints having a straightforward interpretation, making the model much more parsimonious. Estimation based on the maximum likelihood (ML) approach is developed under different constraints based on the Expectation-Maximization algorithm. Steps of Newton-Raphson type are also included to improve the algorithm’s convergence speed