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75f( ˆX|Y, ˆφ, ˆθ) ∝ ∏ if(Y i | ˆX i , ˆθ)p( ˆX i |N( ˆX i ), ˆφ) (5.11)In some situations, one might wish to conduct inference based on this pseudoposterior.However, we use the ICM reconstruction of X, so our only remaininggoal is to conduct inference for K, the number of segments in the image.5.1.4 Pseudolikelihood and BICWe wish to conduct inference for K, the number of segments in the image. Aninitial thought would be to use BIC, as discussed in section 3.1. However, thiswould require evaluation of the likelihood of the observed data, L(Y |K), which isshown in equation 5.12.L(Y |K) = ∑ xf(Y |X = x, K)p(X = x|K) (5.12)The sum in equation 5.12 involves all possible configurations of the hiddenstates. With N pixels and K states, there are K N possible configurations, makingthis approach intractable. Instead, we replace the likelihood term with a pseudolikelihoodwhich maintains computational feasibility.An alternative to this pseudolikelihood method is to explore the space of allpossible configurations of hidden states. This can be done using reversible jumpMarkov chain Monte Carlo (Green, 1995), which would yield an estimate of theposterior probability of each K value. The main drawback of reversible jumpMCMC is its large computational demand. In most cases, we will not really beinterested in the posterior probabilities of values of K; instead, we simply wanta single best K value. In the context of the consistency result shown below,we would expect that as the amount of data increases, the choice of the singlebest K should be the same for both the reversible jump MCMC method and thepseudolikelihood-based BIC method.