Consistent order estimation for nonparametric Hidden Markov Models

TL;DR

This paper introduces two novel methods for consistently estimating the number of hidden states in nonparametric HMMs, achieving theoretical guarantees without restrictive assumptions and demonstrating practical effectiveness through experiments.

Contribution

It provides the first consistency proofs for order estimation in general nonparametric HMMs without prior bounds or parametric constraints, and introduces two new estimation techniques.

Findings

Both methods are proven to be almost surely consistent.

The first method yields rate minimax adaptive estimators.

Numerical experiments show effective order selection in various scenarios.

Abstract

We consider the problem of estimating the number of hidden states (the order) of a nonparametric hidden Markov model (HMM). We propose two different methods and prove their almost sure consistency without any prior assumption, be it on the order or on the emission distributions. This is the first time a consistency result is proved in such a general setting without using restrictive assumptions such as a priori upper bounds on the order or parametric restrictions on the emission distributions. Our main method relies on the minimization of a penalized least squares criterion. In addition to the consistency of the order estimation, we also prove that this method yields rate minimax adaptive estimators of the parameters of the HMM - up to a logarithmic factor. Our second method relies on estimating the rank of a matrix obtained from the distribution of two consecutive observations. Finally, numerical experiments are used to compare both methods and study their ability to select the right order in several situations.