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74Once each pixel has been updated, we have a new ˆX. We can now check forconvergence of ˆX, or stop after a predetermined number of iterations. At the endof ICM, we also have ˆθ and ˆφ. I now comment on the posterior distribution of X,which might be used for inference in some applications, before I continue on to adiscussion of inference for the number of segments K.5.1.3 Pseudoposterior Distribution of the True SceneThis section presents a pseudolikelihood-based expression for the posterior distributionof the true segmentation X, which is the distribution of X conditional onthe observed Y . Recall that we do not observe the X values, but we do observeY i for each pixel. We assume that the density of Y i conditional on its true stateX i = j is Gaussian with mean µ j and variance σj 2 , and it follows that the Y valuesare conditionally independent given the X values.f(X|Y ) = f(Y |X)f(X)/f(Y ) ∝ f(Y |X)f(X) (5.8)The first term is easy to compute, since f(Y |X) = ∏ i f(Y i |X i ), and this isjust a Gaussian density. Here we replace the second term by the pseudolikelihoodP L(X). Dependence on the parameter φ is made explicit in the equations whichfollow.P L(X, φ) = ∏ ip(X i |N(X i ), φ) (5.9)p(X i |N(X i ), φ) = exp(φU(N(X i), X i ))∑k exp(φU(N(X i ), k))(5.10)Rewriting this in computational terms, we obtain the following expression forthe pseudoposterior distribution of X, based on the pseudolikelihood.