introduction to statistical decision theory

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INTRODUCTION TO STATISTICAL DECISION THEORY John W. Pratt, Howard Raiffa, and Robert Schlaifer The MIT Press Cambridge, Massachusetts London, England

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Page 1: INTRODUCTION TO STATISTICAL DECISION THEORY

INTRODUCTION TO STATISTICAL DECISION THEORY

John W. Pratt, Howard Raiffa, and Robert Schlaifer

The MIT Press Cambridge, Massachusetts London, England

Page 2: INTRODUCTION TO STATISTICAL DECISION THEORY

Contents

Preface xv

Introduction 1 1.1 The Problem of Decision under Uncertainty 1 1.2 Decision Trees 3 1.3 The Problem of Analysis 6

An Informal Treatment of Foundations 11 2.1 Introduction 11 2.2 Canonical Probability 12 2.3 Basic Assumptions 14 2.4 Comparison of Simple Canonical Lotteries 19 2.5 Comparison of Simple "Real" Lotteries: An Introduction 24 2.6 Consistency Requirements for Evaluations of Events 26 2.7 Reduction of Acts to Reference Lotteries 32 2.8 Conditional Utility and Conditional Probability 36 Exercises 42

A Formal Treatment of Foundations 47 3.1 Introduction 47 3.2 The Canonical Basis 49 3.3 Lotteries and Prizes 51 3.4 Formal Notation and Definitions 54 3.5 Axioms or Basic Assumptions 55 3.6 Results 59 3.7 Conditional Probability 64

Assessment of Utilities for Consequences 69 4.1 Indifference Probabilities and Utility Indices 69 4.2 Utility Functions for Monetary Consequences 71 4.3 Construction of a Utility Function for Money 77 4.4 Monetary and Other Numeraires 81 Exercises 83 Case 85

Quantification of Judgments 93 5.1 Introduction 93 5.2 Consistency of a Set of Probability Assessments 93 5.3 Relative Frequency and the Rational Assessment of Probabilities 96 5.4 Abstract Probability 103

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5.5 Results on Partitions, Double Partitions, and Two-Way Tables 107 Exercises 109

Analysis of Decision Trees 113 6.1 Description of a Class of Decision Problems 113 6.2 The General Decision Problem as a Game 114 6.3 Illustrative Examples 114 6.4 Analysis of a Decision Tree 124 Exercises 129

Random Variables 133 7.1 Introduction: Definition of a Random Variable 133 7.2 Discrete Random Variables: Mass Functions 138 7.3 Continuous Random Variables: Density Functions 143 7.4 Mixed Random Variables 148 7.5 Functions of Random Variables 148 7.6 Probability Assessments for a RV 150 Exercises 155

Continuous Lotteries and Expectations 159 8.1 Reduction of Lotteries with an Infinity of Consequences 159 8.2 Expectations 167 8.3 Variance of a Random Variable 172 8.4 Functions Defined as Expectations 174 8.5 Reduction of Lotteries Using the Expectation Operator 176 Exercises 177

Special Univariate Distributions 181 9.1 Introduction 181 9.2 Mathematical Preliminaries: Complete Beta and Gamma Functions 181 9.3 The Beta Distribution 182 9.4 The Binomial Distribution 186 9.5 The Pascal Distribution 188 9.6 The Hyperbinomial, Hyperpascal, and Hypergeometric Distributions 189 9.7 The Normal Distribution 192 9.8 The Gamma-2 Distribution 195 9.9 The Student Distribution 197 9.10 The Exponential Distribution 200 9.11 The Gamma-1 Distribution 202

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Contents

9.12 The Poisson Distribution 205 9.13 The Negative-Binomial Distribution 206 Exercises 208

Conditional Probability and Bayes' Theorem 211 10.1 Introduction 211 10.2 Conditional Preferences for Lotteries 211 10.3 Bayes' Theorem 218 10.4 Summary of Main Results of Section 10.2 221

Bernoulli Process 225 11.1 The Bernoulli Model 225 11.2 Probability Assignments for a Bernoulli Process with Known p 225 11.3 Probability Assignments for a Bernoulli Process with Unknown p 226 11.4 The Beta Family of Priors 231 11.5 Selection of a Distribution to Express Judgments about p 237 Exercises 241

Terminal Analysis: Opportunity Loss and the Value of Perfect Information 247 12.1 Introduction 247 12.2 Opportunity Loss and the Value of Perfect Information 247 12.3 Two-Action Problems with Linear Value 253 12.4 Finite-Action Problems with Linear Value 256 12.5 Point Estimation 258 12.6 Infinite-Action Problems with Quadratic Loss 260 12.7 Infinite-Action Problems with Linear Loss 261 12.8 Classification with a Zero-One Loss Structure 264 12.9 Comparison of Summary Measures 265 Exercises 265

Paired Random Variables 273 13.1 Introduction: Definition of a Paired Random Variable 273 13.2 Discrete Paired Random Variables 274 13.3 Mixed Paired Random Variables 279 13.4 Continuous Paired Random Variables 284 13.5 Independence 287 13.6 Indirect Assessment of Joint Distributions 288 13.7 Expectations 290 13.8 Mean, Variance, Covariance, and Correlation 295 Exercises 301

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Contents

Preposterior Analysis: The Value of Sample Information 307 14.1 Introduction 307 14.2 Basic Assumptions: Linear Preference; Additive Terminal and Sampling Values 307 14.3 The General Method of Analysis 308 14.4 Sampling from a Bernoulli Process 312 14.5 The Infinite-Action Problem with Quadratic Loss 317 14.6 The Infinite-Action Problem with Linear Loss 322 14.7 The Two-Action Problem with Linear Value 323 14.8 Sequential Sampling 327 Exercises 331 Cases 338

Poisson Process 345 15.1 The Poisson Model with Known Parameter 345 15.2 Conditional Sampling Distributions 354 15.3 Posterior Distribution of I 356 15.4 The Gamma-1 Family of Priors 357 15.5 Sensitivity of Posterior Distributions to Prior Parameters of the Family 358 15.6 Preposterior Distribution Theory 360 15.7 Terminal and Preposterior Analysis 364 15.8 The Infinite-Action Problem with Quadratic Loss 365 15.9 The Infinite-Action Problem with Linear Loss 366 15.10 The Two-Action Problem with Linear Loss 368 15.11 Selection of a Distribution to Express Judgments about I 369 15.12 Use of the Poisson Process as an Approximation to a Bernoulli Process with Small p 371

Normal Process with Known Variance 375 16.1 The Normal Data-Generating Process 375 16.2 Conditional Sampling Distributions 378 16.3 Posterior Distribution of ß 380 16.4 Posterior Analysis When the Prior Distribution of ß is Normal 382 16.5 Sensitivity of the Posterior to the Prior 385 16.6 Preposterior Distribution Theory 387 16.7 Terminal and Preposterior Analyses 394 16.8 Infinite-Action Problems with Quadratic Loss 396 16.9 The Infinite-Action Problem with Linear Loss 397 16.10 The Two-Action Problem with Linear Value 399

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16.11 Effect of Nonoptimal Sample Size 410 Exercises 412

17 Normal Process with Unknown Variance 417 17.1 Introduction 417 17.2 Conditional Sampling Distributions 418 17.3 Posterior Distribution of (ß, v) 420 17.4 Posterior Analysis When the Joint Prior Distribution of (ß, v) is Normal-Inverted-Gamma 420 17.5 Sensitivity of the Posterior to the Prior 423 17.6 Preposterior Distribution Theory 426 17.7 Terminal and Preposterior Analysis 429 17.8 Infinite-Action Problems with Quadratic Loss 430 17.9 Alternate Approaches 432

18 Large Sample Theory 437 18.1 The Central Limit Theorem 437 18.2 Normal Approximation to Mathematically Well-Defined Distributions 447 18.3 Introduction to Large Sample Theory: On the Intractability of Multiparameter Processes 451 18.4 Use of the Sample Mean as a Summary Statistic 454

19 Statistical Analysis in Normal Form 463 19.1 Comparison of Extensive-Form and Normal-Form Analyses 463 19.2 Infinite-Action Problems 467 19.3 Two-Action Problems with Breakeven Values 484 Exercises 495 Appendix: Statistical Decision Theory from on Objectivistic Viewpoint 503

20 Classical Methods 517 20.1 Models and "Objective" Probabilities 517 20.2 Point Estimation 519 20.3 Confidence Intervals 522 20.4 Testing Hypotheses 529 20.5 Tests of Significance as Sequential Decision Procedures 541 20.6 The Likelihood Principle and Optional Stopping 542 20.7 Further Uses of Tests of Hypotheses 545 Appendix: Outline of Some Aspects of Sufficient Statistics 546

21 Multivariate Random Variables 551 21.1 Introduction: Definition of a Multivariate Random Variable 551

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21.2 Discrete Multivariate RV's 553 21.3 Mixed Multivariate Random Variables 560 21.4 Continuous Multivariate RV's 560 21.5 Independence 563 21.6 Indirect Assessment of Joint Distributions 568 21.7 Expectations 570 21.8 Mean and Variance of a Vector RV 571 21.9 Linear Transformations: Characteristics of a Variance Matrix 575 Exercises 578

22 The Multivariate Normal Distribution 585 22.1 The Unit Spherical Multivariate Normal Distribution 585 22.2 The General Nonsingular Multivariate Normal Distribution 589 22.3 Marginal and Conditional Distributions 594 22.4 Linear Transformations and Singular Distributions 605 22.5 Some Comments on the Assessment of Multivariate Priors 611 Exercises 623

23A Choosing the Best of Several Processes 639 23.1 Introduction, Definitions, and Assumptions 639 23.2 Posterior Analysis 640 23.3 Values as a RV; Terminal Analysis 641 23.4 Expected Value of Perfect Information 641 23.5 The Distribution of ? and EVSI 646 23.6 A Numerical Example 649 Exercises 651

23B Allowance for Uncertain Bias 655 23.7 The Problem of Bias 655 23.8 Probability Assumptions 658 23.9 Posterior Distributions 661 23.10 Preposterior Distributions 662 23.11 Economic Analysis, Biased Sampling Only 662 23.12 Economic Analysis, Biased and Unbiased Sampling 669 Exercises 676 Cases 680

23C Stratification 689 23.13 Introduction: Stratified Sampling 689 23.14 Probability Assumptions 690 23.15 Posterior Analysis and Terminal Decisions 691

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23.16 Outline of the Rest of the Chapter 692 23.17 Preposterior Analysis 692 23.18 Nonoptimal Allocation (Diffuse Prior) 699 23.19 Stratified Random Sampling versus Simple Random Sampling 703 Exercises 710

23D The Portfolio Problem 713 23.20 Introduction 713 23.21 Available Means and Standard Deviations 715 23.22 Utility Determined by Mean and Standard Deviation 717 23.23 Finding Points on the Efficient Boundary 721 23.24 Extensions 726 Exercises 728

24 Normal Linear Regression with Known Variance 731 24.1 Definition of the Model 731 24.2 "Gentle" Prior 737 24.3 Normal Prior 742 24.4 Special Cases 746 24.5 Not All xs Known in Advance 750 24.6 Choice among Models 754 24.7 Causation 757 Exercises 768

Appendix 1: The Terminology of Sets 781 Al.l The General Idea of a Set 781 AI.2 Relations among Sets 781 A1.3 Special Types of Sets: Notation 783

Appendix 2: Elements of Matrix Theory 785 A l l Definition ofa Matrix 785 A2.2 Operations on Matrices 785 A2.3 Inverse of a Matrix 787 A2.4 Partitioned Matrices 788 A2.5 Transformations and Inverses 789 A2.6 Linear Combinations of Vectors, Rank of a Matrix 791 A2.7 Positive-Definite and Positive-Semidefinite Matrices and Quadratic Forms 792 A2.8 Geometrical Representation of Vectors and Quadratic Forms 794 A2.9 Determinants 796

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A2.10 Inner Products, Distance, and Orthogonality 798 A2.ll Canonical Representation of Quadratic Forms 799 A2.12 Characteristic Roots and Vectors 801 Exercises 803

Appendix 3: Properties of Utility Functions for Monetary Consequences 805 A3.1 Introduction 805 A3.2 Measuring Risk Aversion 805 A3.3 Examples 806 A3.4 Decreasing Risk Aversion 807 A3.5 More Risk Averse 808 A3.6 Allocations to a Single Risk 808 A3.7 Independent Risks 809 A3.8 Further Comments on the Choice of a Utility Function 810 Exercises 812

Appendix 4: Tables 819

Bibliography 861 Index 865