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This time, however, the problem can be attributed primarily to the classifier. A cursory inspection of Fig. 5 reveals that both our assumptions of statistical independence and normality can be questioned, and our assumption that the marginal densities p ( x , I B) and p ( x , 1 8) differed only in their mean values is particularly bad. A somewhat better assumption, one that at least takes account of the different scale factors for x1 and x 2 , is to assume that 20 R. 0. DUDA where u12 is the variance of x1 and uZ2is the variance of x 2 .

40) can be recursively solved by setting the terminal condition to be and computing backwards for risk functions R , , n < N. T h e major 50 K. S. FU difference between the solution of Eq. 40) and that of Eq.

48) c assigning x to w1 if g ( x ) > 0 and to augmented vectors a and y by a = x) w2 if g ( x ) I:[ < 0. 50) we can write g(x) in the homogeneous form ‘The problem of designing such a classifier is the problem of finding an (augmented) weight vector a from a set of sample patterns. T h e following procedure due to Rosenblatt (1957) is typical of one class of adaptive procedures for solving this problem. , having the property that each sample appears infinitely often in the sequence. Let x,, be arbitrary, and let I n words, after y k is observed, ak is either increased by y k , decreased by y, , or left unchanged.

### A density lemma

by Robert

4.2