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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 {\displaystyle Y_{i}} , where i ∈ S {\displaystyle i\in S}. Hidden Markov random fields assume that the probabilistic nature of Y i {\displaystyle Y_{i}} is determined by the unobservable Markov random field X i {\displaystyle X_{i}} , i ∈ S {\displaystyle i\in S}.That is, given the neighbors N i {\displaystyle N_{i}} of X i , X i {\displaystyle X_{i},X_{i}} is independent of all other X j {\displaystyle X_{j}} .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 {\displaystyle 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 {\displaystyle X_{i}} , Y i {\displaystyle Y_{i}} are independent.
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.