A Probabilistic Theory of Pattern Recognition by Luc Devroye

By Luc Devroye

Pattern popularity provides probably the most major demanding situations for scientists and engineers, and plenty of diverse techniques were proposed. the purpose of this booklet is to supply a self-contained account of probabilistic research of those ways. The booklet features a dialogue of distance measures, nonparametric tools in accordance with kernels or nearest pals, Vapnik-Chervonenkis concept, epsilon entropy, parametric category, errors estimation, unfastened classifiers, and neural networks. anywhere attainable, distribution-free homes and inequalities are derived. a considerable section of the consequences or the research is new. Over 430 difficulties and workouts supplement the material.

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Other quantities have been suggested over the years that measure the discriminatory power hidden in the distribution of (X, Y). These may be helpful in some settings. For example, in theoretical studies or in certain proofs, the relationship between L *and the distribution of (X, Y) may become clearer via certain inequalities that link L * with other functionals of the distribution. Perhaps we may even learn a thing or two about what it is that makes L * small. In feature selection, some explicit inequalities involving L *may provide just the kind of numerical information that will allow one to make certain judgements on what kind of feature is preferable in practice.

The relationship between p and L * is not linear though. We will show that for all distributions, LNN is more useful than p if it is to be used as an approximation of L *. 1. For all distributions, we have I 1 --J14p 2 2 2 < 1 1 -2L 2 -JI2 < L* :'S p. NN PROOF. First of all, E 2 { J11(X)(I - 17(X))} p2 < E [17(X)(l - 17(X))} (by Jensen's inequality) 24 3. 1)). J7J(l - 7]) ~ 27](1 - 7J)for all7J E [0, 1], we see that p ~ LNN ~ L *. Finally, by the Cover-Hart inequality, J1-2LNN ~ JI-4L*(I- L*)= l-2L*.

In fact, then, all discrimination problems are one-dimensional, as we could equally well replace X by I)( X) or by any strictly monotone function of I)(X), such as 1) 7 (X) + 51)\X) + I)(X). Unfortunately, 1) is unknown in general. -T-Bl) and in another case 1)(T) =max (0, 1 - (1 + 7- T)e- 0 -Tl). The former format suggests that we could base all decisions on T + B. This means that if we had no access to T and B individually, but to T + B jointly, we would be able to achieve the same results! Since 1J is unknown, all ofthis is really irrelevant.

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