ISBN-10:
1584886226
ISBN-13:
9781584886228
Pub. Date:
08/19/2010
Publisher:
Taylor & Francis
Utility-Based Learning from Data / Edition 1

Utility-Based Learning from Data / Edition 1

by Craig Friedman, Sven SandowCraig Friedman
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Overview

Utility-Based Learning from Data provides a pedagogical, self-contained discussion of probability estimation methods via a coherent approach from the viewpoint of a decision maker who acts in an uncertain environment. This approach is motivated by the idea that probabilistic models are usually not learned for their own sake; rather, they are used to make decisions. Specifically, the authors adopt the point of view of a decision maker who


(i) operates in an uncertain environment where the consequences of every possible outcome are explicitly monetized,
(ii) bases his decisions on a probabilistic model, and
(iii) builds and assesses his models accordingly.

These assumptions are naturally expressed in the language of utility theory, which is well known from finance and decision theory. By taking this point of view, the book sheds light on and generalizes some popular statistical learning approaches, connecting ideas from information theory, statistics, and finance. It strikes a balance between rigor and intuition, conveying the main ideas to as wide an audience as possible.

Product Details

ISBN-13: 9781584886228
Publisher: Taylor & Francis
Publication date: 08/19/2010
Series: Chapman & Hall/CRC Machine Learning & Pattern Recognition
Pages: 417
Product dimensions: 6.20(w) x 9.40(h) x 1.10(d)

About the Author

Craig Friedman is a managing director and head of research in the Quantitative Analytics group at Standard & Poor’s in New York. Dr. Friedman is also a fellow of New York University’s Courant Institute of Mathematical Sciences. He is an associate editor of both the International Journal of Theoretical and Applied Finance and the Journal of Credit Risk.

Sven Sandow is an executive director in risk management at Morgan Stanley in New York. Dr. Sandow is also a fellow of New York University’s Courant Institute of Mathematical Sciences. He holds a Ph.D. in physics and has published articles in scientific journals on various topics in physics, finance, statistics, and machine learning.

The contents of this book are Dr. Sandow’s opinions and do not represent Morgan Stanley.

Table of Contents

Preface xv

Acknowledgments xvii

Disclaimer xix

1 Introduction 1

1.1 Notions from Utility Theory 2

1.2 Model Performance Measurement 4

1.2.1 Complete versus Incomplete Markets 7

1.2.2 Logarithmic Utility 7

1.3 Model Estimation 8

1.3.1 Review of Some Information-Theoretic Approaches 8

1.3.2 Approach Based on the Model Performance Measurement Principle of Section 1.2 12

1.3.3 Information-Theoretic Approaches Revisited 15

1.3.4 Complete versus Incomplete Markets 16

1.3.5 A Data-Consistency Tuning Principle 17

1.3.6 A Summary Diagram for This Model Estimation, Given a Set of Data-Consistency Constraints 18

1.3.7 Problem Settings in Finance, Traditional Statistical Modeling, and This Book 18

1.4 The Viewpoint of This Book 20

1.5 Organization of This Book 21

1.6 Examples 22

2 Mathematical Preliminaries 33

2.1 Some Probabilistic Concepts 33

2.1.1 Probability Space 33

2.1.2 Random Variables 35

2.1.3 Probability Distributions 35

2.1.4 Univariate Transformations of Random Variables 40

2.1.5 Multivariate Transformations of Random Variables 41

2.1.6 Expectations 42

2.1.7 Some Inequalities 43

2.1.8 Joint, Marginal, and Conditional Probabilities 44

2.1.9 Conditional Expectations 45

2.1.10 Convergence 46

2.1.11 Limit Theorems 48

2.1.12 Gaussian Distributions 48

2.2 Convex Optimization 50

2.2.1 Convex Sets and Convex Functions 50

2.2.2 Convex Conjugate Function 52

2.2.3 Local and Global Minima 53

2.2.4 Convex Optimization Problem 54

2.2.5 Dual Problem 54

2.2.6 Complementary Slackness and Karush-Kuhn-Tucker (KKT) Conditions 57

2.2.7 Lagrange Parameters and Sensitivities 57

2.2.8 Minimax Theorems 58

2.2.9 Relaxation of Equality Constraints 59

2.2.10 Proofs for Section 2.2.9 62

2.3 Entropy and Relative Entropy 63

2.3.1 Entropy for Unconditional Probabilities on Discrete State Spaces 64

2.3.2 Relative Entropy for Unconditional Probabilities on Discrete State Spaces 67

2.3.3 Conditional Entropy and Relative Entropy 69

2.3.4 Mutual Information and Channel Capacity Theorem 70

2.3.5 Entropy and Relative Entropy for Probability Densities 71

2.4 Exercises 73

3 The Horse Race 79

3.1 The Basic Idea of an Investor in a Horse Race 80

3.2 The Expected Wealth Growth Rate 81

3.3 The Kelly Investor 82

3.4 Entropy and Wealth Growth Rate 83

3.5 The Conditional Horse Race 85

3.6 Exercises 92

4 Elements of Utility Theory 95

4.1 Beginnings: The St. Petersburg Paradox 95

4.2 Axiomatic Approach 98

4.2.1 Utility of Wealth 102

4.3 Risk Aversion 102

4.4 Some Popular Utility Functions 104

4.5 Field Studies 106

4.6 Our Assumptions 106

4.6.1 Blowup and Saturation 107

4.7 Exercises 108

5 The Horse Race and Utility 111

5.1 The Discrete Unconditional Horse Races 111

5.1.1 Compatibility 111

5.1.2 Allocation 114

5.1.3 Horse Races with Homogeneous Returns 118

5.1.4 The Kelly Investor Revisited 119

5.1.5 Generalized Logarithmic Utility Function 120

5.1.6 The Power Utility 122

5.2 Discrete Conditional Horse Races 123

5.2.1 Compatibility 123

5.2.2 Allocation 125

5.2.3 Generalized Logarithmic Utility Function 126

5.3 Continuous Unconditional Horse Races 126

5.3.1 The Discretization and the Limiting Expected Utility 126

5.3.2 Compatibility 128

5.3.3 Allocation 130

5.3.4 Connection with Discrete Random Variables 132

5.4 Continuous Conditional Horse Races 133

5.4.1 Compatibility 133

5.4.2 Allocation 135

5.4.3 Generalized Logarithmic Utility Function 137

5.5 Exercises 137

6 Select Methods for Measuring Model Performance 139

6.1 Rank-Based Methods for Two-State Models 139

6.2 Likelihood 144

6.2.1 Definition of Likelihood 145

6.2.2 Likelihood Principle 145

6.2.3 Likelihood Ratio and Neyman-Pearson Lemma 149

6.2.4 Likelihood and Horse Race 150

6.2.5 Likelihood for Conditional Probabilities and Probability Densities 151

6.3 Performance Measurement via Loss Function 152

6.4 Exercises 153

7 A Utility-Based Approach to Information Theory 155

7.1 Interpreting Entropy and Relative Entropy in the Discrete Horse Race Context 156

7.2 (U, O)-Entropy and Relative (U, O)-Entropy for Discrete Unconditional Probabilities 157

7.2.1 Connection with Kullback-Leibler Relative Entropy 158

7.2.2 Properties of (U, O)-Entropy and Relative (U, O)-Entropy 159

7.2.3 Characterization of Expected Utility under Model Mis-specification 162

7.2.4 A Useful Information-Theoretic Quantity 163

7.3 Conditional (U, O)-Entropy and Conditional Relative (U, O)-Entropy for Discrete Probabilities 163

7.4 U-Entropy for Discrete Unconditional Probabilities 165

7.4.1 Definitions of U-Entropy and Relative U-Entropy 166

7.4.2 Properties of U-Entropy and Relative U-Entropy 168

7.4.3 Power Utility 176

7.5 Exercises 179

8 Utility-Based Model Performance Measurement 181

8.1 Utility-Based Performance Measures for Discrete Probability Models 183

8.1.1 The Power Utility 185

8.1.2 The Kelly Investor 186

8.1.3 Horse Races with Homogeneous Returns 186

8.1.4 Generalized Logarithmic Utility Function and the Log-Likelihood Ratio 187

8.1.5 Approximating the Relative Model Performance Measure with the Log-Likelihood Ratio 189

8.1.6 Odds Ratio Independent Relative Performance Measure 190

8.1.7 A Numerical Example 191

8.2 Revisiting the Likelihood Ratio 192

8.3 Utility-Based Performance Measures for Discrete Conditional Probability Models 194

8.3.1 The Conditional Kelly Investor 196

8.3.2 Generalized Logarithmic Utility Function, Likelihood Ratio, and Odds Ratio Independent Relative Performance Measure 196

8.4 Utility-Based Performance Measures for Probability Density Models 198

8.4.1 Performance Measures and Properties 198

8.5 Utility-Based Performance Measures for Conditional Probability Density Models 198

8.6 Monetary Value of a Model Upgrade 199

8.6.1 General Idea and Definition of Model Value 200

8.6.2 Relationship between V and Δ 201

8.6.3 Best Upgrade Value 201

8.6.4 Investors with Power Utility Functions 202

8.6.5 Approximating V for Nearly Homogeneous Expected Returns 203

8.6.6 Investors with Generalized Logarithmic Utility Functions 204

8.6.7 The Example from Section 8.1.7 205

8.6.8 Extension to Conditional Probabilities 205

8.7 Some Proofs 207

8.7.1 Proof of Theorem 8.3 207

8.7.2 Proof of Theorem 8.4 209

8.7.3 Proof of Theorem 8.5 214

8.7.4 Proof of Theorem 8.10 220

8.7.5 Proof of Corollary 8.2 and Corollary 8.3 221

8.7.6 Proof of Theorem 8.11 223

8.8 Exercises 226

9 Select Methods for Estimating Probabilistic Models 229

9.1 Classical Parametric Methods 230

9.1.1 General Idea 230

9.1.2 Properties of Parameter Estimators 231

9.1.3 Maximum-Likelihood Inference 234

9.2 Regularized Maximum-Likelihood Inference 236

9.2.1 Regularization and Feature Selection 238

9.2.2 lκ-Regularization, the Ridge, and the Lasso 239

9.3 Bayesian Inference 240

9.3.1 Prior and Posterior Measures 240

9.3.2 Prior and Posterior Predictive Measures 242

9.3.3 Asymptotic Analysis 243

9.3.4 Posterior Maximum and the Maximum-Likelihood Method 246

9.4 Minimum Relative Entropy (MRE) Methods 248

9.4.1 Standard MRE Problem 249

9.4.2 Relation of MRE to ME and MMI 250

9.4.3 Relaxed MRE 250

9.4.4 Proof of Theorem 9.1 254

9.5 Exercises 255

10 A Utility-Based Approach to Probability Estimation 259

10.1 Discrete Probability Models 262

10.1.1 The Robust Outperformance Principle 263

10.1.2 The Minimum Market Exploitability Principle 267

10.1.3 Minimum Relative (U, O)-Entropy Modeling 269

10.1.4 An Efficient Frontier Formulation 271

10.1.5 Dual Problem 278

10.1.6 Utilities Admitting Odds Ratio Independent Problems: A Logarithmic Family 285

10.1.7 A Summary Diagram 286

10.2 Conditional Density Models 286

10.2.1 Preliminaries 288

10.2.2 Modeling Approach 290

10.2.3 Dual Problem 292

10.2.4 Summary of Modeling Approach 297

10.3 Probability Estimation via Relative U-Entropy Minimization 297

10.4 Expressing the Data Constraints in Purely Economic Terms 301

10.5 Some Proofs 303

10.5.1 Proof of Lemma 10.2 303

10.5.2 Proof of Theorem 10.3 303

10.5.3 Dual Problem for the Generalized Logarithmic Utility 308

10.5.4 Dual Problem for the Conditional Density Model 309

10.6 Exercises 310

11 Extensions 313

11.1 Model Performance Measures and MRE for Leveraged Investors 313

11.1.1 The Leveraged Investor in a Horse Race 313

11.1.2 Optimal Betting Weights 314

11.1.3 Performance Measure 316

11.1.4 Generalized Logarithmic Utility Functions: Likelihood Ratio as Performance Measure 317

11.1.5 All Utilities That Lead to Odds-Ratio Independent Relative Performance Measures 318

11.1.6 Relative (U, O)-Entropy and Model Learning 318

11.1.7 Proof of Theorem 11.1 318

11.2 Model Performance Measures and MRE for Investors in Incomplete Markets 320

11.2.1 Investors in Incomplete Markets 320

11.2.2 Relative U-Entropy 324

11.2.3 Model Performance Measure 327

11.2.4 Model Value 331

11.2.5 Minimum Relative U-Entropy Modeling 332

11.2.6 Proof of Theorem 11.6 334

11.3 Utility-Based Performance Measures for Regression Models 334

11.3.1 Regression Models 336

11.3.2 Utility-Based Performance Measures 337

11.3.3 Robust Allocation and Relative (U, O)-Entropy 338

11.3.4 Performance Measure for Investors with a Generalized Logarithmic Utility Function 340

11.3.5 Dual of Problem 11.2 347

12 Select Applications 349

12.1 Three Credit Risk Models 349

12.1.1 A One-Year Horizon Private Firm Default Probability Model 351

12.1.2 A Debt Recovery Model 356

12.1.3 Single Period Conditional Ratings Transition Probabilities 363

12.2 The Gail Breast Cancer Model 370

12.2.1 Attribute Selection and Relative Risk Estimation 371

12.2.2 Baseline Age-Specific Incidence Rate Estimation 372

12.2.3 Long-Term Probabilities 373

12.3 A Text Classification Model 374

12.3.1 Datasets 374

12.3.2 Term Weights 375

12.3.3 Models 376

12.4 A Fat-Tailed, Flexible, Asset Return Model 377

References 379

Index 391

What People are Saying About This

From the Publisher

Utility-Based Learning from Data is an excellent treatment of data-driven statistics for decision-making. Friedman and Sandow lucidly describe the connections between different branches of statistics and econometrics, such as utility theory, maximum entropy, and Bayesian analysis. A must-read for serious statisticians!
—Marco Avellaneda, Professor of Mathematics, New York University, and Risk Magazine Quant of the Year 2010

Combining insights from both theory and practice, this is a model trade book about modeling trading books.
—Peter Carr, Global Head of Market Modeling, Morgan Stanley, and Executive Director, Masters in Math Finance, New York University

Utility-Based Learning from Data connects key ideas from utility theory with methods from statistics, machine learning, and information theory. It presents, using decision-theoretic principles, a framework for building models that can be used by decision makers. By adopting the utility-based approach, Friedman and Sandow are able to adapt models to the risk preferences of the model user, while maintaining tractability. It is a much-needed and comprehensive book, which should help put model-building for use by decision makers on more solid ground.
—Gregory Piatetsky-Shapiro, editor of KDnuggets.com, co-founder and past Chair of SIGKDD, and founder of the Knowledge Discovery and Data Mining (KDD) conferences

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