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78The consistency result presented here is shown for a limited case, but it ishoped that future work might extend the result to more general cases. I first statethe theorem, and then two lemmas precede the proof.Theorem 5.1: Consistency of Choice of KWe observe an image Y consisting of pixels Y i which we assume are each generatedfrom a Gaussian distribution which depends on the true state X i of each pixel. Wedefine i to index only the interior pixels of the image. We assume that the localcharacteristics of the true state image X can be modeled as a Markov random field;in particular, we assume the Potts model given by equation 5.1. Let K T denote thetrue number of segments in the image, and let K denote a hypothesized numberof segments.Suppose that one of the following cases holds.Case 1: K T = 1 and K > 1.Case 2: K T = 2, K = 1, and condition A: log(σ K ) − log(σ 1 ) − 8φ > 0, whereσ K is the standard deviation from the K = 1 fit and σ 1 largest of the two standarddeviations from the K T = 2 fit.In case 1 or case 2, BIC P L (K) is consistent for K; that is, as N → ∞ in such away that the size of the image increases in both dimensions,P KT (BIC P L (K) < BIC P L (K T )) → 1 (5.17)Condition AThis condition is relevant only when K T = 2 and K = 1. Denote the true densityparameters in θ K T by µ1 , µ 2 , σ1, 2 and σ2. 2 Similarly, let the parameters in θ K be