Inference in Mixed Hidden Markov Models and Applications to Medical Studies

The aim of the present paper is to document the need for adapting the definition of hidden Markov models (HMM) to population studies, as well as for corresponding learning methodologies. In this article, mixed hidden Markov models (MHMM) are introduced through a brief state of the art on hidden Markov models and related applications, especially focusing on disease related problems. Making the main assumption that a given pathology can be considered at different stages, hidden Markov models have for example already been used to study epileptic activity or migraine. Mixed-effects hidden Markov models have been newly introduced in the statistical literature. The notion of mixed hidden Markov models is particularly relevant for modeling medical symptoms, but the data complexity generally requires specific care and the available methodology for MHMM is relatively poor. Our new approach can be briefly described as follows. First, we suggest to estimate the population parameters with the SAEM (Stochastic Approximation EM) algorithm, which has the property to converge quickly. The well-known forward recursions developed for HMM allow to compute easily the complete likelihood at each step of the MCMC procedure used within SAEM. Then, for dealing with the individuals, we suggest to estimate each set of individual parameters with the MAP (Maximum A Posteriori) of the parameter distributions. Finally, the hidden state sequences are decoded using the Viterbi algorithm. Some Monte-Carlo experiments are presented to illustrate the accuracy of our algorithms.