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76The basic idea of this psuedolikelihood approach is that instead of summingover all possible configurations of X we will consider only configurations whichare close to the ICM estimate of X, denoted by ˆX. Specifically, we consider eachpixel Y i in turn and condition on ˆX −i , which is ˆX excluding the value at X i . Thelikelihood of the ith pixel observation is L(Y i |K), shown in equation 5.13K∑L(Y i |K) = f(Y i |X i = j)p(X i = j) (5.13)j=1The sum in equation 5.13 is now over the K possible values of X i . Conditioningon ˆX −i , we obtain the conditional likelihood shown in equation 5.14, in whichN( ˆX i ) denotes the neighbors of ˆXi .K∑L(Y i | ˆX −i , K) = f(Y i |X i = j)p(X i = j|N( ˆX i )) (5.14)j=1The first term in the sum, f(Y i |X i = j), simply requires evaluation of a Gaussiandensity; the second term, p(X i = j|N( ˆX i )) is evaluated using equation 5.2.The conditional likelihoods from equation 5.14 are combined to form the pseudolikelihoodof the image, L ˆX(Y |K), shown in equation 5.15. Forming the productin this way makes intuitive sense because the Y i values are independent conditionalon the underlying hidden states.L ˆX (Y |K) = ∏ if(Y i | ˆX −i , ˆφ) = ∏ iK∑f(Y i |X i = j)p(X i = j|N( ˆX i ), ˆφ) (5.15)j=1Recall that Y i is the ith observed pixel, X i is the hidden state of pixel i, ˆφ is theestimate of φ (from the ICM algorithm), and N( ˆX i ) is the estimate of the hiddenstate of each neighbor of pixel i. I use L ˆX(Y |K) to denote the quantity in equation5.15, since it is a likelihood integrated over the approximate posterior distributionof a set of models near the MAP estimate ˆX.