Judith Rousseau (Université Paris Dauphine)

Behaviour of the posterior distribution in HMM models when the number of states is misspecified

In this paper we study the asymptotic behaviour of the posterior distribution for parametric HMM models with finite number of components. We concentrate in particular on the case where the number of states of the hidden Markov chain in the model is larger than the true number of states. In such cases the parameter is not identifiable. We study the impact of the prior on the transition matrix on the asymptotic behaviour of the posterior distribution. In particular that if some independent Dirichlet priors are considered on each row of the transition matrix we can identify different types of behaviour of the asymptotic posterior distribution depending on the choice of the parameter of the Dirichlet distributions. A link is made with static mixture models and some numerical study is conducted to illustrate the results and investigate the behaviour of the posterior in cases where no theoretical results are provided.