In this chapter we develop a generalization of hidden Markov models valid for the evolution of a system in continuous time. That is, we describe how to model and analyze hidden Markov jump processes. In this context, we introduce the concept of uniformization to simulate continuous time trajectories and then use uniformization to develop a Monte Carlo strategy to sample trajectories from a posterior over trajectories. Having discussed how trajectories can be sampled from the posterior over all candidate trajectories, we then describe strategies for full posterior inference over trajectories and other model parameters. We end with strategies for trajectory marginalization and continuous time filtering.

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.