Hidden Markov models (HMMs) are flexible tools for clustering dependent data coming from unknown populations, allowing nonparametric identifiability of the population densities when the data is « truly » dependent. In the first part of the talk, I will talk about our result characterizing the frontier between learnable and unlearnable two-state nonparametric HMMs in term of a suitable notion of « distance » to independence. I will present surprising new phenomena emerging in the nonparametric setting. In particular, it is possible to « borrow strength » from the estimator of the smoothest density to improve the estimation of the other.  We conduct a precise analysis of minimax rates, showing a transition depending on compared smoothnesses of the emission densities. In the (short) second part, I will present our result on clustering HMM data. We compute upper and lower bounds on the Bayes risk of clustering and we identify the key quantity determining the difficulty of the clustering task.

First part is joint work with Kweku Abraham and Elisabeth Gassiat, second part is joint work with Elisabeth Gassiat and Ibrahim Kaddouri.