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. 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 iin 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 iin 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}} (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 {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 (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.