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38the image with this independence assumption is given in equation 3.16.N∑ K∑L(Y |K, θ) = log( P j Φ(Y i |θ j )) (3.16)i=1 j=1However, pixels are typically not independent. Chapter 4 explores the impactof autoregressive dependence on the BIC in both one-dimensional and twodimensionaldata, and presents a combined mixture model and raster-scan autoregression(RSA) model which allows for autoregressive dependence with themixture model. In chapter 5, I use a Markov random field model for the spatialdependence.3.3 BIC with Mixture ModelsWe wish to use BIC for model selection with mixture models which have differentnumbers of components. BIC can be computed by using the likelihood fromequation 3.16 in the BIC formula from equation 3.12. The resulting BIC formulais shown in equation 3.17.BIC(K) = 2L(Y |K) − D K log(N) (3.17)Recall from section 3.1 that the BIC approximation is based on the use ofLaplace’s method to approximate the integrated likelihood in a Bayes factor. Aregularity condition for Laplace’s method is that the parameters must be in theinterior of the parameter space; as they approach the boundary of parameter space,the approximation breaks down. This presents a problem for the use of BIC inthe mixture model context. Suppose we fit a model with K segments when thetrue number of segments K 0 is smaller than K. In this case, the true mixtureproportion (P j in equation 3.15) for each extra segment would be zero, which is atthe boundary of parameter space. Also, there would be no information available