If the theory of Bayesian approaches in standard nonparametric or high dimensional models is beginning to be well developed, not so much is known in the context of semi-parametric models outside very specific priors and models. We propose in this talk a pseudo Bayesian approach, based on the cut posterior which allows for the construction of a distribution on the whole parameter and is constructed such that the marginal posterior on the parameter of interest has optimal properties. We apply this approach to the setup of nonparametric hidden Markov models with finite state space and nonparametric emission distributions.
Since the seminal paper of Gassiat et al. (2016), it is known that in such models the transition matrix \(Q\) and the emission distributions \(F_1,\dots, F_K\) are identifiable, up to label switching. We cut a posterior to simultaneously estimate \(Q\) at the rate \(\sqrt{n}\) and the emission distributions at the usual nonparametric rates. To do so, we first consider a prior \(\pi_1\) on \(Q\) and \(F_1,\dots, F_K\) which leads to a posterior marginal distribution on \(Q\) which verifies the Bernstein von Mises property and thus to an estimator of \(Q\) which is efficient. We then combine the marginal posterior on \(Q\) with an other posterior distribution on the emission distributions, following the cut-posterior approach, to obtain a posterior which also concentrates around the emission distributions at the minimax rates. In addition an important intermediate result of our work is an inversion inequality which allows to upper bound the \(L_1\) norms between the emission densities by the \(L_1\) norms between marginal densities of 3 consecutive observations.