Mixtures of Nonparametric Components and Hidden Markov Models

The topic of this chapter is statistical inference of nonparametric finite mixtures. The latent variables (and thus the observations) will be mostly taken independent and identically distributed, but in some cases, they will be possibly non independently distributed. For each observation, the corresponding latent variable indicates from which population the observation comes from. In particular, when the latent variables form a Markov chain, the observation process will comme from a non parametric hidden Markov model (HMM) with finite state space. We would like to emphasise the fact that the nonparametric modeling will concern only the conditional distribution of the observations, conditional on the latent variables, not the mixing distribution. Nonparametric modeling of the mixing distribution (with possibly infinitely denumerable or continuous support) is considered in Chapter 6. To fix ideas, assume that a random variable X follows a distribution