multivariate poisson hidden markov models for analysis of spatial ...

According to the local characteristics of MRFs, the joint probability of any pair of
( X( s), Y( s )), given X ( s ) ’s neighborhood configuration X ( N ), is
p( y(), s x()| s x( Ν )) = p( y()| s x()) s p( x()| s x( Ν )).
s
The marginal probability distribution of Y ( s ) dependent on the parameter set θ and
X ( Ν ) can be written as
s
py ( ( s) | x( Νs), θ) = ∑ py ( ( s), l| x( Νs), θ)
∈L
= ∑ f( y( s); θ
) p( | x( Νs))
∈L
where = { θ : ∈ L}
s
θ
.
This model is called the hidden Markov random field model. Note that the concept of an
HMRF is different from that of an MRF in the sense that the former is defined with
respect to a pair of random variable families, ( X , Y)
while the latter is only defined
with respect to X .
s
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