In this chapter we introduce the clustering problem and use it to motivate mixture models. We start by describing clustering in a frequentist paradigm and introduce the relevant likelihoods and latent variables. We then discuss properties of the likelihoods including invariance with respect to label swapping. Finally, we expand this discussion to describe clustering and mixture models more generally within a Bayesian paradigm. This allows us to introduce Dirichlet priors used in inferring the weight we ascribe to each cluster component from which the data are drawn. Finally, we describe the infinite mixture model and Dirichlet process priors within the Bayesian nonparametric paradigm, appropriate for the analysis of uncharacterized data that may contain an unspecified number of clusters.

In this chapter we introduce and apply hidden Markov models to model and analyze dynamical data. Hidden Markov models are one of simplest of dynamical models valid for systems evolving in a discrete state-space at discrete time points. We first describe the evaluation of the likelihood relevant to hidden Markov models and introduce the concept of filtering. We then describe how to obtain maximum likelihood estimators using expectation maximization. We then broaden our discussion to the Bayesian paradigm and introduce the Bayesian hidden Markov model. In this context, we describe the forward filtering backward sampling algorithm and Monte Carlo methods for sampling from hidden Markov model posteriors. As hidden Markov models are flexible modeling tools, we present a number of variants including the sticky hidden Markov model, the factorial hidden Markov model, and the infinite hidden Markov model. Finally, we conclude with a case study in fluorescence spectroscopy where we show how the basic filtering theory presented earlier may be extended to evaluate the likelihood of a second-order hidden Markov model.