Non Parametric Hidden Markov Models with Finite State Space: Posterior Concentration Rates

Abstract

The use of non parametric hidden Markov models with finite state space is flourishing in practice while few theoretical guarantees are known in this framework. Here, we study asymptotic guarantees for these models in the Bayesian framework. We obtain posterior concentration rates with respect to the $L_1$-norm on joint marginal densities of consecutive observations in a general theorem. We apply this theorem to two cases and obtain minimax concentration rates. We consider discrete observations with emission distributions distributed from a Dirichlet process and continuous observations with emission distributions distributed from Dirichlet process mixtures of Gaussian distributions.

References in corpus (7)

• Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
• A Bernstein-Von Mises Theorem for discrete probability distributions
• About the posterior distribution in hidden Markov models with unknown number of states
• Minimax adaptive estimation of nonparametric hidden Markov models
• A Nonparametric Bayesian Approach to Uncovering Rat Hippocampal Population Codes During Spatial Navigation
• Minimax Estimation of Discrete Distributions under ${\ell }_1$ Loss
• Nonparametric identification and maximum likelihood estimation for hidden Markov model

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