Nonasymptotic control of the MLE for misspecified nonparametric hidden Markov models

2018/07/11 by Luc Lehéricy, Lehéricy, Luc
Computer Science · Engineering · #Bayesian Methods and Mixture Models #FOS: Mathematics #Fault Detection and Control Systems #Machine Learning and Algorithms #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1807.03997

openalex publication_date 2018/07/11 · openalex created_date 2022/11/10 · openalex updated_date 2026/07/28

Abstract

Finite state space hidden Markov models are flexible tools to model phenomena with complex time dependencies: any process distribution can be approximated by a hidden Markov model with enough hidden states.We consider the problem of estimating an unknown process distribution using nonparametric hidden Markov models in the misspecified setting, that is when the data-generating process may not be a hidden Markov model.We show that when the true distribution is exponentially mixing and satisfies a forgetting assumption, the maximum likelihood estimator recovers the best approximation of the true distribution. We prove a finite sample bound on the resulting error and show that it is optimal in the minimax sense--up to logarithmic factors--when the model is well specified.

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