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 formula_1, where formula_2. Hidden Markov random fields assume that the probabilistic nature of formula_1 is determined by the unobservable Markov random field formula_4, formula_2. That is, given the neighbors formula_6 of formula_7 is independent of all other formula_8 (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. formula_4 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 formula_4, formula_1 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.