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Chapter 3MARGINAL SEGMENTATIONIn this chapter I present a discussion of image segmentation based on themarginal (without spatial information) pixel values. I begin by presenting somebackground on the Bayesian Information Criterion (BIC), and then I introduce themixture model. I discuss the use of BIC for model selection in this context, andI examine two schemes for classification of pixels after model selection has beendone.3.1 BICThe Bayesian Information Criterion (BIC) was first given by Schwarz (1978) in thecontext of model selection with IID observations from a certain class of densities,and Haughton (1988) extended the class of densities to curved exponential families.The difference in the BIC value between two models is an approximation of twicethe log of the Bayes factor comparing the two models; the BIC has the advantageof being relatively easy to compute. The basic idea of the BIC is to use Laplace’smethod to approximate the integrated likelihood in the Bayes factor, and thenignore terms which do not increase quickly with N. In this section I present aderivation of the BIC, which largely follows the discussions found in Kass andRaftery (1995) and Raftery (1995). The following sections show how the BIC canbe adjusted for use with the AR(1) model and for the raster scan autoregression(RSA) model.A common approach to comparing two models or hypotheses, say M 2 vs. M 1 ,