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Lower bounds for Bayes error estimation

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3 Author(s)
A. Antos ; Comput. & Autom. Res. Inst., Hungarian Acad. of Sci., Budapest, Hungary ; L. Devroye ; L. Gyorfi

We give a short proof of the following result. Let (X,Y) be any distribution on N×{0,1}, and let (X1,Y1),...,(Xn,Yn) be an i.i.d. sample drawn from this distribution. In discrimination, the Bayes error L*=infgP{g(X)≠Y} is of crucial importance. Here we show that without further conditions on the distribution of (X,Y), no rate-of-convergence results can be obtained. Let φn(X1,Y1,...,Xn,Yn ) be an estimate of the Bayes error, and let {φn(.)} be a sequence of such estimates. For any sequence {an} of positive numbers converging to zero, a distribution of (X,Y) may be found such that E{|L*-φn(X1,Y 1,...,Xn,Yn)|}⩾an often converges infinitely

Published in:

IEEE Transactions on Pattern Analysis and Machine Intelligence  (Volume:21 ,  Issue: 7 )