Hidden Markov models

Thus far, we have worked with time series, but here we formally introduce their treatment. A time series consists of data measured in succession in time. The treatment of time series is a vast subject, and we will eventually deal with some of the many ways to model them, including filtering and smoothing.

Here, we introduce the concept of a hidden Markov model, or HMM. In its simplest incarnation, which is what we will study, the observation at each point in time is dependent on a latent variable at that point in time. The latent variable is itself dependent on the value of the latent variable at the previous point in time. The latent variables take on a finite discrete set of states over time.

As an example, one may think of the mood of a subject as a latent variable and neuronal activity as the observation.