A Probabilistic Theory of Pattern Recognition by Luc Devroye

By Luc Devroye

Pattern acceptance provides probably the most major demanding situations for scientists and engineers, and plenty of diverse ways were proposed. the purpose of this booklet is to supply a self-contained account of probabilistic research of those techniques. The ebook features a dialogue of distance measures, nonparametric tools in accordance with kernels or nearest buddies, Vapnik-Chervonenkis idea, epsilon entropy, parametric class, blunders estimation, loose classifiers, and neural networks. anyplace attainable, distribution-free houses and inequalities are derived. a considerable element of the implications or the research is new. Over 430 difficulties and workouts supplement the material.

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Assume that the components of X = (X (I 1, ... , x

C. 3. 2 are illustrated here. FIGURE £:::;; 1-l(L*,l-L*) 0 log 2 0 PROOF. PART A. Define A £ =min(7](X), 1 - 7](X)). Then = E{'H(A, 1 -A)} < 1-l(EA, I - EA) (because 1t is concave, by Jensen's inequality) = PART 1-l(L*,l- L*). B. 6 Jeffreys' Divergence = - log(l > 27 LNN) - log(l - L *). PART C. 1)) = log2-~(1-2L*) 2 • o REMARK. The nearly monotone relationship between £ and L * will see lots of uses. We warn the reader that near the origin, L * may decrease linearly in£, but it may also decrease much faster than £ 209 • Such wide variation was not observed in the relationship between L * and LNN (where it is linear) or L * and p (where it is between linear and quadratic).

Take (x', y') = ( -oo, I). Then the probability of error is p. Clearly, PROOF. S: min(p, I- p). This proves the first part of the lemma. For the second part, if L = 1/2, then p = 1/2, and for every x, pF1(x) + (1 - p)(l - F0 (x)) 2: 1/2 and p(1 - F,(x)) + (1- p) F0 (x) 2: I /2. S: p- 1/2. Therefore, L = 1/2 means that for every x, pF1(x) - (I - p)F0 (x) = p - 1/2. Thus, for all x, F 1(x) = F0 (x ), and therefore L * = I /2. 2. I· 2 In particular, 2 X if p = I /2. then I I 2 2 L =---sup IF,(x)- Fo(x)l.

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