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44In finding argmax C g(C|X), we can now consider each pixel separately sinceeach term in the sum over i only involves pixel i. Let N j denote the number ofpixels in class j. The maximization takes the following form.K ( )∑ 1C i = argmax m I(X i , j)I(X i , m) (3.28)N jj=1Because of the two indicator functions, terms in the sum will be nonzero onlywhen j = m. Note that I(X i , m) 2 = I(X i , m).C i = argmax m( 1N m)I(X i , m) (3.29)We can now return to our modeling assumptions and multiply by N N .Note that N N m=( )1 NC i = argmax m P (X i = m|Y i ) (3.30)N N m1P (X i, and apply Bayes’ theorem as in equation 3.23.=m)()1 1C i = argmax m P (Y i |X i = m)P (X i = m) (3.31)N P (X i = m)C i = argmax m P (Y i |X i = m) (3.32)As shown in equation 3.15, we have modeled the distribution of Y i , so we cansubstitute into equation 3.32 to obtain equation 3.33.C i = argmax m Φ(Y i |θ m ) (3.33)End of Proof.Application of equation 3.33 means that we should classify each pixel into itsmost likely component, without regard for the mixture proportion of each com-