An Elementary Introduction to Statistical

A joint endeavor from leading researchers in the fields of philosophy and electrical engineering, An Introduction to Statistical Learning Theory provides a broad and accessible introduction to rapidly evolving field of statistical pattern recognition and statistical learning theory. Exploring topics that are not often covered in introductory level books on statistical learning theory, including PAC learning, VC dimension, and simplicity, the authors present upper-undergraduate and graduate levels with the basic theory behind contemporary machine learning and uniquely suggest it serves as an excellent framework for philosophical thinking about inductive inference.

Sanjeev Kulkarni, PhD, is Professor in the Department of Electrical Engineering at Princeton University, where he is also an affiliated faculty member in the Department of Operations Research and Financial Engineering and the Department of Philosophy. Dr. Kulkarni has published widely on statistical pattern recognition, nonparametric estimation, machine learning, information theory, and other areas. A Fellow of the IEEE, he was awarded Princeton University's President's Award for Distinguished Teaching in 2007. Gilbert Harman, Phd, is James S. McDonnell Distinguished University Professor in the Department of Philosophy at Princeton University. A Fellow of the Cognitive Science Society, he is the author of more than fifty published articles in his areas of research interest, which include ethics, statistical learning theory, psychology of reasoning, and logic.

Preface	p. xiii
Introduction: Classification, Learning, Features and Applications	p. 1
Scope	p. 1
Why Machine Learning?	p. 2
Some Applications	p. 3
Image Recognition	p. 3
Speech Recognition	p. 3
Medical Diagnosis	p. 4
Statistical Arbitrage	p. 4
Measurements, Features, and Feature Vectors	p. 4
The Need for Probability	p. 5
Supervised Learning	p. 5
Summary	p. 6
Appendix: Induction	p. 6
Questions	p. 7
References	p. 8
Probability	p. 10
Probability of Some Basic Events	p. 10
Probabilities of Compound Events	p. 12
Conditional Probability	p. 13
Drawing Without Replacement	p. 14
A Classic Birthday Problem	p. 15
Random Variables	p. 15
Expected Value	p. 16
Variance	p. 17
Summary	p. 19
Appendix: Interpretations of Probability	p. 19
Questions	p. 20
References	p. 21
Probability Densities	p. 23
An Example in Two Dimensions	p. 23
Random Numbers in [0,1]	p. 23
Density Functions	p. 24
Probability Densities in Higher Dimensions	p. 27
Joint and Conditional Densities	p. 28
Expected Value and Variance	p. 28
Laws of Large Numbers	p. 29
Summary	p. 30
Appendix: Measurability	p. 30
Question	p. 32
References	p. 32
The Pattern Recognition Problem	p. 34
A Simple Example	p. 34
Decision Rules	p. 35
Success Criterion	p. 37
The Best Classifier: Bayes Decision Rule	p. 37
Continuous Features and Densities	p. 38
Summary	p. 39
Appendix: Uncountably Many	p. 39
Questions	p. 40
References	p. 41
The Optimal Bayes Decision Rule	p. 43
Bayes Theorem	p. 43
Bayes Decision Rule	p. 44
Optimality and Some Comments	p. 45
An Example	p. 47
Bayes Theorem and Decision Rule with Densities	p. 48
Summary	p. 49
Appendix: Defining Conditional Probability	p. 50
Questions	p. 50
References	p. 53
Learning from Examples	p. 55
Lack of Knowledge of Distributions	p. 55
Training Data	p. 56
Assumptions on the Training Data	p. 57
A Brute Force Approach to Learning	p. 59
Curse of Dimensionality, Inductive Bias, and No Free Lunch	p. 60
Summary	p. 61
Appendix: What Sort of Learning?	p. 62
Questions	p. 63
References	p. 64
The Nearest Neighbor Rule	p. 65
The Nearest Neighbor Rule	p. 65
Performance of the Nearest Neighbor Rule	p. 66
Intuition and Proof Sketch of Performance	p. 67
Using more Neighbors	p. 69
Summary	p. 70
Appendix: When People use Nearest Neighbor Reasoning	p. 70
Who Is a Bachelor?	p. 70
Legal Reasoning	p. 71
Moral Reasoning	p. 71
Questions	p. 72
References	p. 73
Kernel Rules	p. 74
Motivation	p. 74
A Variation on Nearest Neighbor Rules	p. 75
Kernel Rules	p. 76
Universal Consistency of Kernel Rules	p. 79
Potential Functions	p. 80
More General Kernels	p. 81
Summary	p. 82
Appendix: Kernels, Similarity, and Features	p. 82
Questions	p. 83
References	p. 84
Neural Networks: Perceptrons	p. 86
Multilayer Feedforward Networks	p. 86
Neural Networks for Learning and Classification	p. 87
Perceptrons	p. 89
Threshold	p. 90
Learning Rule for Perceptrons	p. 90
Representional Capabilities of Perceptrons	p. 92
Summary	p. 94
Appendix: Models of Mind	p. 95
Questions	p. 96
References	p. 97
Multilayer Networks	p. 99
Representation Capabilities of Multilayer Networks	p. 99
Learning and Sigmoidal Outputs	p. 101
Training Error and Weight Space	p. 104
Error Minimization by Gradient Descent	p. 105
Backpropagation	p. 106
Derivation of Backpropagation Equations	p. 109
Derivation for a Single Unit	p. 110
Derivation for a Network	p. 111
Summary	p. 113
Appendix: Gradient Descent and Reasoning toward Reflective Equilibrium	p. 113
Questions	p. 114
References	p. 115
PAC Learning	p. 116
Class of Decision Rules	p. 117
Best Rule from a Class	p. 118
Probably Approximately Correct Criterion	p. 119
PAC Learning	p. 120
Summary	p. 122
Appendix: Identifying Indiscernibles	p. 122
Questions	p. 123
References	p. 123
VC Dimension	p. 125
Approximation and Estimation Errors	p. 125
Shattering	p. 126
VC Dimension	p. 127
Learning Result	p. 128
Some Examples	p. 129
Application to Neural Nets	p. 132
Summary	p. 133
Appendix: VC Dimension and Popper Dimension	p. 133
Questions	p. 134
References	p. 135
Infinite VC Dimension	p. 137
A Hierarchy of Classes and Modified PAC Criterion	p. 138
Misfit Versus Complexity Trade-Off	p. 138
Learning Results	p. 139
Inductive Bias and Simplicity	p. 140
Summary	p. 141
Appendix: Uniform Convergence and Universal Consistency	p. 141
Questions	p. 142
References	p. 143
The Function Estimation Problem	p. 144
Estimation	p. 144
Success Criterion	p. 145
Best Estimator: Regression Function	p. 146
Summary	p. 147
Appendix: Regression Toward the Mean	p. 147
Questions	p. 148
References	p. 149
Learning Function Estimation	p. 150
Review of the Function Estimation/Regression Problem	p. 150
Nearest Neighbor Rules	p. 151
Kernel Methods	p. 151
Neural Network Learning	p. 152
Estimation with a Fixed Class of Functions	p. 153
Shattering, Pseudo-Dimension, and Learning	p. 154
Conclusion	p. 156
Appendix: Accuracy, Precision, Bias, and Variance in Estimation	p. 156
Questions	p. 157
References	p. 158
Simplicity	p. 160
Simplicity in Science	p. 160
Explicit Appeals to Simplicity	p. 160
Is the World Simple?	p. 161
Mistaken Appeals to Simplicity	p. 161
Implicit Appeals to Simplicity	p. 161
Ordering Hypotheses	p. 162
Two Kinds of Simplicity Orderings	p. 162
Two Examples	p. 163
Curve Fitting	p. 163
Enumerative Induction	p. 164
Simplicity as Simplicity of Representation	p. 165
Fix on a Particular System of Representation?	p. 166
Are Fewer Parameters Simpler?	p. 167
Pragmatic Theory of Simplicity	p. 167
Simplicity and Global Indeterminacy	p. 168
Summary	p. 169
Appendix: Basic Science and Statistical Learning Theory	p. 169
Questions	p. 170
	p. 170
Support Vector Machines	p. 172
Mapping the Feature Vectors	p. 173
Maximizing the Margin	p. 175
Optimization and Support Vectors	p. 177
Implementation and Connection to Kernel Methods	p. 179
Details of the Optimization Problem	p. 180
Rewriting Separation Conditions	p. 180
Equation for Margin	p. 181
Slack Variables for Nonseparable Examples	p. 181
Reformulation and Solution of Optimization	p. 182
Summary	p. 183
Appendix: Computation	p. 184
Questions	p. 185
References	p. 186
Boosting	p. 187
Weak Learning Rules	p. 187
Combining Classifiers	p. 188
Distribution on the Training Examples	p. 189
The Adaboost Algorithm	p. 190
Performance on Training Data	p. 191
Generalization Performance	p. 192
Summary	p. 194
Appendix: Ensemble Methods	p. 194
Questions	p. 195
References	p. 196
Bibliography	p. 197
Author Index	p. 203
Subject Index	p. 207
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An Elementary Introduction to Statistical Learning Theory

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