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41The ith pixel of X or C is denoted by subscript. The parameter vector θ, themixture proportion P , and the density Φ are defined in equation 3.15 of section3.2.Proof of Theorem 3.1We define a utility function g(X, C) where X is the true image (i.e. the unobservabletrue classification). Equation 3.19 shows the utility function for mixtureclassification.g(C|X) = (1/N) ∑ iI(X i , C i ) (3.19)Here, I(A, B) = 1 if A = B and 0 otherwise, and X i is the true (unobserved) valueunderlying the observation Y i . We wish to find the value of C to maximize theutility.argmax C g(C|X) = argmax C (1/N) ∑ iI(X i , C i ) (3.20)Since we are not able to observe X, we must restate this in probabilistic terms.Let P (X i = C i |Y i ) denote the probability that X i = C i given that we observe Y i .argmax C g(C|Y ) = argmax C (1/N) ∑ iP (X i = C i |Y i ) (3.21)Inside the sum, each term depends only on a single pixel of C, so maximizationwill be accomplished by choosing as the value of C i that value which maximizesthe probability that X i = C i . Note that X i and C i can take on only the classvalues of 1...K, so the maximization becomesApplying Bayes theorem,C i = argmax m P (X i = m|Y i ) (3.22)