Hidden Markov random field

In statistics, a hidden Markov random field is a generalization of a hidden Markov model. Instead of having an underlying Markov chain, hidden Markov random fields have an underlying Markov random field.

Suppose that we observe a random variable $$ Y_i $$, where $$ i \in S $$. Hidden Markov random fields assume that the probabilistic nature of $$ Y_i $$ is determined by the unobservable Markov random field $$ X_i $$, $$ i \in S $$. That is, given the neighbors $$ N_i $$ of $$ X_i, X_i $$ is independent of all other $$ X_j $$ (Markov property). The main difference with a hidden Markov model is that neighborhood is not defined in 1 dimension but within a network, i.e. $$ X_i $$ is allowed to have more than the two neighbors that it would have in a Markov chain. The model is formulated in such a way that given $$ X_i $$, $$ Y_i $$ are independent (conditional independence of the observable variables given the Markov random field).

In the vast majority of the related literature, the number of possible latent states is considered a user-defined constant. However, ideas from nonparametric Bayesian statistics, which allow for data-driven inference of the number of states, have been also recently investigated with success, e.g.