Nonasymptotic control of the MLE for misspecified nonparametric hidden Markov models

We study the problem of estimating an unknown time process distribution using nonparametric hidden Markov models in the misspecified setting, that is when the true distribution of the process may not come from 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.