Context Tree Estimation in Variable Length Hidden Markov Models

We address the issue of context tree estimation in variable length hidden Markov models. We propose an estimator of the context tree of the hidden Markov process, which needs no prior upper bound on the depth of the context tree. We prove that the estimator is strongly consistent. This uses information-theoretic mixture inequalities in the spirit of the literatures of Finesso, and Gassiat and Boucheron. We propose an algorithm to efficiently compute the estimator and provide simulation studies to support our result.