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
∈
S
. Hidden Markov random fields assume that the probabilistic nature of
Y
i
is determined by the unobservable Markov random field
X
i
,
i
∈
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).
