Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for Pr {S – ES ≥ nt} depend only on the endpoints of the ranges of the summands and the mean, or the mean and the variance of S. These results are then used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population.

/pdf/probability-inequalities-for-sums-of-bounded-random-1jzyi7tbo3.pdf

Probability Inequalities for sums of Bounded Random Variables

Preface * Introduction * The Bayes Error * Inequalities and alternatedistance measures * Linear discrimination * Nearest neighbor rules *Consistency * Slow rates of convergence Error estimation * The regularhistogram rule * Kernel rules Consistency of the k-nearest neighborrule * Vapnik-Chervonenkis theory * Combinatorial aspects of Vapnik-Chervonenkis theory * Lower bounds for empirical classifier selection* The maximum likelihood principle * Parametric classification *Generalized linear discrimination * Complexity regularization *Condensed and edited nearest neighbor rules * Tree classifiers * Data-dependent partitioning * Splitting the data * The resubstitutionestimate * Deleted estimates of the error probability * Automatickernel rules * Automatic nearest neighbor rules * Hypercubes anddiscrete spaces * Epsilon entropy and totally bounded sets * Uniformlaws of large numbers * Neural networks * Other error estimates *Feature extraction * Appendix * Notation * References * Index

/pdf/a-probabilistic-theory-of-pattern-recognition-1p19dg485x.pdf

A Probabilistic Theory of Pattern Recognition

If S is a random variable with finite rnean and variance, the Bienayme-Chebyshev inequality states that for x > 0, 

$$\Pr \left[ {\left| {S - ES} \right| \geqslant x{{{(\operatorname{var} S)}}^{{1/2}}}} \right] \leqslant {{x}^{{ - 2}}}$$

(1)

If S is the surn of n independent, identically distributed random variables, then, by the central limit theorem*, as n → ∞, the probability on the left approaehes 2Ф( - x), where Ф(x) is the standard normal distribution function. For x large, Ф( - x) behaves as const. x -1 exp( - x2/2).

Probability Inequalities for Sums of Bounded Random Variables

This graduate-level textbook introduces fundamental concepts and methods in machine learning. It describes several important modern algorithms, provides the theoretical underpinnings of these algorithms, and illustrates key aspects for their application. The authors aim to present novel theoretical tools and concepts while giving concise proofs even for relatively advanced topics. Foundations of Machine Learning fills the need for a general textbook that also offers theoretical details and an emphasis on proofs. Certain topics that are often treated with insufficient attention are discussed in more detail here; for example, entire chapters are devoted to regression, multi-class classification, and ranking. The first three chapters lay the theoretical foundation for what follows, but each remaining chapter is mostly self-contained. The appendix offers a concise probability review, a short introduction to convex optimization, tools for concentration bounds, and several basic properties of matrices and norms used in the book. The book is intended for graduate students and researchers in machine learning, statistics, and related areas; it can be used either as a textbook or as a reference text for a research seminar.

Foundations of Machine Learning

The dip test measures multimodality in a sample by the maximum difference, over all sample points, between the empirical distribution function, and the unimodal distribution function that minimizes that maximum difference. The uniform distribution is the asymptotically least favorable unimodal distribution, and the distribution of the test statistic is determined asymptotically and empirically when sampling from the uniform.

/pdf/the-dip-test-of-unimodality-35m8rstuzx.pdf

The Dip Test of Unimodality

This paper proves a number of inequalities which improve on existing upper limits to the probability distribution of the sum of independent random variables. The inequalities presented require knowledge only of the variance of the sum and the means and bounds of the component random variables. They are applicable when the number of component random variables is small and/or have different distributions. Figures show the improvement on existing inequalities.

https://www.jstor.org/stable/pdfplus/2282438.pdf

Probability Inequalities for the Sum of Independent Random Variables

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Probability Inequalities for the Sum of Independent Random Variables