The Bayes decision functions dj (x ) = p (x/ωj )P (ωj ), j = 1, 2,...,W, were derived using a 0-1 loss function. Prove that these decision functions minimize the probability of error. (Hint: The probability of error p (e )is 1 − p (c ),where p (c )is the probability of being correct. For a pattern vector x belonging to class ωi, p (c/x ) = p(ωi/x ). Find p (c )and show that p (c )is maximum [p (e )is minimum] when p(x/ωi)P(ωi) is maximum.)
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