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77After running ICM, we have an estimate of φ, as well as estimates of the µand σ parameters for each segment. We can compute the log of the quantity inequation 5.15, and, since this is an approximation of the intractable L(Y |K), weuse it in place of the loglikelihood in the BIC, as shown in equation 5.16. I usethe notation BIC P L (K) to differentiate this equation from the usual BIC.BIC P L (K) = 2 log(L ˆX (Y |K)) − D K log(N) (5.16)Ideally, one could compute BIC P L (K) for a large range of K values and thenchoose K to maximize BIC P L (K). However, this would require an excessiveamount of computation, and we do not expect the model assumptions to holdfor values of K very far from the true value. Because of this, we adopt a sequentialtesting approach. We begin by computing BIC P L (K) for K = 1, andthen incrementally increase the value of K. At each step, we compare BIC P L (K)with BIC P L (K − 1), and stop the process when the larger model is rejected. Inother words, as we increase K incrementally from K = 1, we take the first localmaximum of BIC P L (K) to be our choice of the number of segments K.5.1.5 Consistency of BIC P LEquation 5.16 gives the formula for BIC P L (K) which I use for model selection.In this section I present a consistency result for BIC P L (K). First, I refine thenotation. Let K T denote the true number of segments. Let ˆX K be the estimatedX given that there are K segments, and similarly let ˆθ K and ˆφ K be the parameterestimates given K segments. N(X i ) is the neighborhood of X i , which consists ofthe 8 pixels adjacent to X i , and B i is the union of X i and its neighborhood. Inother words, B i is a three by three block of pixels centered at pixel i; omission ofthe subscript will indicate an arbitrary three by three block of pixels. I assumethat f(Y i |X i ) is a Gaussian distribution.