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Any Discrimination Rule Can Have an Arbitrarily Bad Probability of Error for Finite Sample Size

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1 Author(s)
Devroye, L. ; School of Computer Science, McGill University, Montreal, P.Q., Canada.

Consider the basic discrimination problem based on a sample of size n drawn from the distribution of (X, Y) on the Borel sets of Rdx {0, 1}. If 0 < R*. < ¿ is a given number, and ¿n ¿ 0 is an arbitrary positive sequence, then for any discrimination rule one can find a distribution for (X, Y), not depending upon n, with Bayes probability of error R* such that the probability of error (Rn) of the discrimination rule is larger than R* + ¿n for infinitely many n. We give a formal proof of this result, which is a generalization of a result by Cover [1]. Furthermore, sup all distributions of (X, Y) with R* = 0 Rn > ¿. Thus, any attempt to find a nontrivial distribution-free upper bound for Rn will fail, and any results on the rate of convergence of Rn to R* must use assumptions about the distribution of (X, Y).

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Pattern Analysis and Machine Intelligence, IEEE Transactions on  (Volume:PAMI-4 ,  Issue: 2 )