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42P (X i = m|Y i ) ∝ P (Y i |X i = m)P (X i = m) (3.23)As shown in equation 3.15, we have modeled the distribution of Y i , and we canestimate the prior P (X i = m) by using the mixture proportion P m . Substitutinginto equation 3.22,C i = argmax m P m Φ(Y i |θ m ) (3.24)In other words, pixel i is assigned to the component for which it has the highestlikelihood, and we include the mixture proportions (P m ) in the likelihood. Thisseems intuitively reasonable since we are using the mixture density given by equation3.15.End of Proof.Componentwise ClassificationAlthough maximizing the number of correctly classified pixels is both reasonableand consistent with our presumed mixture model, a componentwise approach tothis problem may be more useful in some circumstances. Consider a case in whichwe have a large background and only a few small features of interest. The mixtureproportions will be dominated by the component describing the background, givinginordinate weight to classification of pixels as background. As the proportion ofpixels in the background increases, classification by the mixture likelihood maylead to classifying all pixels as background even when Φ(Y i |θ j ) for the feature isseveral orders of magnitude larger than Φ(Y i |θ j ) for the background. In this case,we would want to use componentwise classification so that the feature componentswould receive a more reasonable share of influence on the classification.