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43In componentwise classification, our aim is to maximize the sum of the proportionsof correctly classified pixels for each component. This gives equal weightto each component. I now derive the classification rule which corresponds to thecomponentwise approach.Theorem 3.2: Optimality of Componentwise ClassificationTo maximize the sum of the proportions of correctly classified pixels for eachcomponent, the optimal classification rule is to assign each pixel to the segmentwhich has the largest component likelihood, as shown in equation 3.25.C i = argmax m Φ(Y i |θ m ) (3.25)Proof of Theorem 3.2To find the appropriate classification rule for the componentwise approach, webegin by stating its utility function.(∑)K∑i I(X i , j)I(X i , C i )g(C|X) =∑j=1i I(X i , j)(3.26)As before, I(A, B) = 1 if A = B and 0 otherwise, and X i is the true (unobserved)value underlying the observation Y i . Note that the term in the denominatorof equation 3.26 is constant with respect to C. The form of equation 3.26is meant to be conceptually clear, but an equivalent and more computationallyuseful form is obtained by moving the denominator into the sum in the numeratorand interchanging the order of summation.g(C|X) = ∑ i⎛ ()⎞K∑ 1⎝ ∑ I(X i , j)I(X i , C i ) ⎠ (3.27)q I(X q , j)j=1