Estimation of Dependences Based on Empirical DataTwenty-?ve years have passed since the publication of the Russian version of the book Estimation of Dependencies Based on Empirical Data (EDBED for short). Twen- ?ve years is a long period of time. During these years many things have happened. Looking back, one can see how rapidly life and technology have changed, and how slow and dif?cult it is to change the theoretical foundation of the technology and its philosophy. I pursued two goals writing this Afterword: to update the technical results presented in EDBED (the easy goal) and to describe a general picture of how the new ideas developed over these years (a much more dif?cult goal). The picture which I would like to present is a very personal (and therefore very biased) account of the development of one particular branch of science, Empirical - ference Science. Such accounts usually are not included in the content of technical publications. I have followed this rule in all of my previous books. But this time I would like to violate it for the following reasons. First of all, for me EDBED is the important milestone in the development of empirical inference theory and I would like to explain why. S- ond, during these years, there were a lot of discussions between supporters of the new 1 paradigm (now it is called the VC theory ) and the old one (classical statistics). |
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Sisältö
| 1 | |
| 8 | |
| 15 | |
| 22 | |
| 39 | |
A Generalization of the GlivenkoCantelli Theorem and the Problem of Pattern Recognition | 41 |
Remarks on Two Procedures for Minimizing Expected Risk on the Basis of Empirical Data | 42 |
Methods of Parametric Statistics for the Pattern Recognition Problem | 45 |
A Deterministic Statement of the Problem | 144 |
Upper Bounds on Error Probabilities | 146 |
An Enet of a Set | 149 |
Necessary and Sufficient Conditions for Uniform Convergence of Frequencies to Probabilities | 152 |
Properties of Growth Functions | 154 |
Bounds on Deviations of Empirically Optimal Decision Rules | 155 |
Remarks on the Bound on the Rate of Uniform Convergence of Frequencies to Probabilities | 158 |
Remark on the General Theory of Uniform Estimating of Probabilities | 159 |
The Pattern Recognition Problem 2 Discriminant Analysis | 46 |
Decision Rules in Problems of Pattern Recognition | 49 |
Evaluation of Qualities of Algorithms for Density Estimation | 51 |
The Bayesian Algorithm for Density Estimation | 52 |
Bayesian Estimators of Discrete Probability Distributions | 54 |
Bayesian Estimators for the Gaussian Normal Density | 56 |
Unbiased Estimators 9 Sufficient Statistics 10 Computing the Best Unbiased Estimator | 66 |
The Problem of Estimating the Parameters of a Density | 70 |
The MaximumLikelihood Method 13 Estimation of Parameters of the Probability Density Using the MaximumLikelihood Method | 73 |
Methods of Parametric Statistics for | 74 |
Problem of Regression Estimation 1 The Scheme for Interpreting the Results of Direct Experiments | 81 |
A Remark on the Statement of the Problem of Interpreting the Results of Direct Experiments | 83 |
Density Models | 84 |
Extremal Properties of Gaussian and Laplace Distributions | 87 |
On Robust Methods of Estimating Location Parameters | 91 |
Robust Estimation of Regression Parameters | 96 |
Robustness of Gaussian and Laplace Distributions | 99 |
Classes of Densities Formed by a Mixture of Densities | 101 |
Densities Concentrated on an Interval | 103 |
Robust Methods for Regression Estimation 101 103 | 105 |
The Problem of Estimating Regression Parameters | 112 |
The Theory of Normal Regression | 113 |
Methods of Estimating the Normal Regression that are Uniformly Superior to the LeastSquares Method | 115 |
A Theorem on Estimating the Mean Vector of a Multivariate Normal Distribution | 120 |
The GaussMarkov Theorem | 125 |
Best Linear Estimators | 127 |
Criteria for the Quality of Estimators | 128 |
Evaluation of the Best Linear Estimators | 130 |
Utilizing Prior Information | 134 |
A Method of Minimizing Empirical Risk for the Problem of Pattern Recognition | 139 |
Uniform Convergence of Frequencies of Events to Their Probabilities | 141 |
A Particular Case | 142 |
Sufficient Conditions | 162 |
A2 The Growth Function | 163 |
A3 The Basic Lemma | 168 |
A4 Derivation of Sufficient Conditions | 170 |
A5 A Bound on the Quantity I | 173 |
A6 A Bound on the Probability of Uniform Relative Deviation | 176 |
A Method of Minimizing Empirical Risk for the Problem of Regression Estimation | 181 |
A Particular Case | 183 |
A Generalization to a Class with Infinitely Many Members | 186 |
The Capacity of a Set of Arbitrary Functions | 188 |
Uniform Boundedness of a Ratio of Moments | 191 |
Two Theorems on Uniform Convergence | 192 |
Theorem on Uniform Relative Deviation | 195 |
Remarks on a General Theory of Risk Estimation | 202 |
Appendix to Chapter 7 Theory of Uniform Convergence | 206 |
A3 Eextension of a Set | 215 |
A7 Corollaries | 231 |
Solution of Illposed Problems Interpretation | 267 |
Appendix to Chapter 9 Statistical Theory of Regularization | 308 |
Estimation of Functional Values at Given Points | 312 |
A Bound on the Uniform Relative Deviation of Means | 318 |
Selection of a Sample for Estimating Values of | 327 |
Estimation of Values of an Indicator Function in the Class | 334 |
The Problem of Finding the Best Point of a Given Set | 341 |
Appendix to Chapter 10 Taxonomy Problems | 347 |
Algorithms for Estimating Nonindicator | 370 |
Bibliographical Remarks | 384 |
Bibliography | 391 |
115 | 396 |
Index | 397 |
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according algorithms approximation arbitrary assertion assume belonging bound called capacity Chapter choose classification close complete computed consider consisting construct contains corresponding decision rules defined definition Denote density dependences determined deviation direction distribution elements empirical risk equality equation equivalence classes errors example exists expected risk expression finite fixed frequencies function F(x given hyperplane idea ill-posed indicator inequality inference lemma linear loss mathematical matrix means measure method metric minimization minimizes the empirical necessary normal Observe obtain operator parameters pattern recognition polynomial possible probability problem PROOF proved quantity regression sample satisfied selected separating sequence side solution solving space splines statistics structure subdivided sufficient takes Theorem theory uniform convergence utilize valid vectors yields
Kirjaluettelon tiedot
| Otsikko | Estimation of Dependences Based on Empirical Data Information Science and Statistics |
| Kirjoittaja | V. Vapnik |
| kääntäjä | S. Kotz |
| Painos | kuvitettu, uusintapainos |
| Kustantaja | Springer Science & Business Media, 2006 |
| ISBN | 0387342397, 9780387342399 |
| Pituus | 505 sivua |
| Vie sitaatti | BiBTeX EndNote RefMan |