Nonasymptotic control of the MLE for misspecified nonparametric hidden Markov models

TL;DR

This paper establishes finite-sample bounds for the maximum likelihood estimator in misspecified nonparametric hidden Markov models, demonstrating near-optimal recovery of the true distribution under certain mixing conditions.

Contribution

It provides the first nonasymptotic analysis of MLE in misspecified nonparametric HMMs, including finite-sample error bounds and minimax optimality results.

Findings

MLE recovers the best approximation of the true distribution.

Finite-sample error bounds are established.

Results are minimax optimal up to logarithmic factors.

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.