INTRODUCTION TO STATISTICAL DECISION THEORY
INTRODUCTION TO STATISTICALDECISION THEORYJohn W. Pratt, Howard Raiffa, and Robert SchlaiferThe MIT PressCambridge, MassachusettsLondon, England
ContentsPrefaceIntroduction1.1 The Problem of Decision under Uncertainty1.2 Decision Trees1.3 The Problem of Analysisxv1136An Informal Treatment of Foundations2.1 Introduction2.2 Canonical Probability2.3 Basic Assumptions2.4 Comparison of Simple Canonical Lotteries2.5 Comparison of Simple "Real" Lotteries: An Introduction2.6 Consistency Requirements for Evaluations of Events2.7 Reduction of Acts to Reference Lotteries2.8 Conditional Utility and Conditional ProbabilityExercises11111214192426323642A Formal Treatment of Foundations3.1 Introduction3.2 The Canonical Basis3.3 Lotteries and Prizes3.4 Formal Notation and Definitions3.5 Axioms or Basic Assumptions3.6 Results3.7 Conditional Probability4747495154555964Assessment of Utilities for Consequences4.1 Indifference Probabilities and Utility Indices4.2 Utility Functions for Monetary Consequences4.3 Construction of a Utility Function for Money4.4 Monetary and Other n of Judgments5.1 Introduction5.2 Consistency of a Set of Probability Assessments5.3 Relative Frequency and the Rational Assessment ofProbabilities5.4 Abstract Probability93939396103
VU1Contents5.5 Results on Partitions, Double Partitions, and Two-WayTablesExercises107109Analysis of Decision Trees6.1 Description of a Class of Decision Problems6.2 The General Decision Problem as a Game6.3 Illustrative Examples6.4 Analysis of a Decision TreeExercises113113114114124129Random Variables7.1 Introduction: Definition of a Random Variable7.2 Discrete Random Variables: Mass Functions7.3 Continuous Random Variables: Density Functions7.4 Mixed Random Variables7.5 Functions of Random Variables7.6 Probability Assessments for a RVExercises133133138143148148150155Continuous Lotteries and Expectations8.1 Reduction of Lotteries with an Infinity of Consequences8.2 Expectations8.3 Variance of a Random Variable8.4 Functions Defined as Expectations8.5 Reduction of Lotteries Using the Expectation OperatorExercises159159167172174176177Special Univariate Distributions9.1 Introduction9.2 Mathematical Preliminaries: Complete Beta and GammaFunctions9.3 The Beta Distribution9.4 The Binomial Distribution9.5 The Pascal Distribution9.6 The Hyperbinomial, Hyperpascal, and HypergeometricDistributions9.7 The Normal Distribution9.8 The Gamma-2 Distribution9.9 The Student Distribution9.10 The Exponential Distribution9.11 The Gamma-1 Distribution181181181182186188189192195197200202
Contents9.12 The Poisson Distribution9.13 The Negative-Binomial DistributionExercises205206208Conditional Probability and Bayes' Theorem10.1 Introduction10.2 Conditional Preferences for Lotteries10.3 Bayes' Theorem10.4 Summary of Main Results of Section 10.2211211211218221Bernoulli Process11.1 The Bernoulli Model11.2 Probability Assignments for a Bernoulli Process withKnown p11.3 Probability Assignments for a Bernoulli Process withUnknown p11.4 The Beta Family of Priors11.5 Selection of a Distribution to Express Judgments about pExercises225225225226231237241Terminal Analysis: Opportunity Loss and the Value of PerfectInformation12.1 Introduction12.2 Opportunity Loss and the Value of Perfect Information12.3 Two-Action Problems with Linear Value12.4 Finite-Action Problems with Linear Value12.5 Point Estimation12.6 Infinite-Action Problems with Quadratic Loss12.7 Infinite-Action Problems with Linear Loss12.8 Classification with a Zero-One Loss Structure12.9 Comparison of Summary Paired Random Variables13.1 Introduction: Definition of a Paired Random Variable13.2 Discrete Paired Random Variables13.3 Mixed Paired Random Variables13.4 Continuous Paired Random Variables13.5 Independence13.6 Indirect Assessment of Joint Distributions13.7 Expectations13.8 Mean, Variance, Covariance, and CorrelationExercises273273274279284287288290295301
ContentsPreposterior Analysis: The Value of Sample Information14.1 Introduction14.2 Basic Assumptions: Linear Preference; Additive Terminaland Sampling Values14.3 The General Method of Analysis14.4 Sampling from a Bernoulli Process14.5 The Infinite-Action Problem with Quadratic Loss14.6 The Infinite-Action Problem with Linear Loss14.7 The Two-Action Problem with Linear Value14.8 Sequential SamplingExercisesCases307307Poisson Process15.1 The Poisson Model with Known Parameter15.2 Conditional Sampling Distributions15.3 Posterior Distribution of I15.4 The Gamma-1 Family of Priors15.5 Sensitivity of Posterior Distributions to Prior Parameters ofthe Family15.6 Preposterior Distribution Theory15.7 Terminal and Preposterior Analysis15.8 The Infinite-Action Problem with Quadratic Loss15.9 The Infinite-Action Problem with Linear Loss15.10 The Two-Action Problem with Linear Loss15.11 Selection of a Distribution to Express Judgments about I15.12 Use of the Poisson Process as an Approximation to aBernoulli Process with Small p345345354356357Normal Process with Known Variance16.1 The Normal Data-Generating Process16.2 Conditional Sampling Distributions16.3 Posterior Distribution of ß16.4 Posterior Analysis When the Prior Distribution of ß isNormal16.5 Sensitivity of the Posterior to the Prior16.6 Preposterior Distribution Theory16.7 Terminal and Preposterior Analyses16.8 Infinite-Action Problems with Quadratic Loss16.9 The Infinite-Action Problem with Linear Loss16.10 The Two-Action Problem with Linear 364365366368369371382385387394396397399
XIContents16.11 Effect of Nonoptimal Sample SizeExercises410412Normal Process with Unknown Variance17.1 Introduction17.2 Conditional Sampling Distributions17.3 Posterior Distribution of (ß, v)17.4 Posterior Analysis When the Joint Prior Distribution of(ß, v) is Normal-Inverted-Gamma17.5 Sensitivity of the Posterior to the Prior17.6 Preposterior Distribution Theory17.7 Terminal and Preposterior Analysis17.8 Infinite-Action Problems with Quadratic Loss17.9 Alternate Approaches417417418420Large Sample Theory18.1 The Central Limit Theorem18.2 Normal Approximation to Mathematically Well-DefinedDistributions18.3 Introduction to Large Sample Theory: On the Intractabilityof Multiparameter Processes18.4 Use of the Sample Mean as a Summary Statistic437437Statistical Analysis in Normal Form19.1 Comparison of Extensive-Form and Normal-Form Analyses19.2 Infinite-Action Problems19.3 Two-Action Problems with Breakeven ValuesExercisesAppendix: Statistical Decision Theory from on ObjectivisticViewpoint46346346748449520Classical Methods20.1 Models and "Objective" Probabilities20.2 Point Estimation20.3 Confidence Intervals20.4 Testing Hypotheses20.5 Tests of Significance as Sequential Decision Procedures20.6 The Likelihood Principle and Optional Stopping20.7 Further Uses of Tests of HypothesesAppendix: Outline of Some Aspects of Sufficient e Random Variables21.1 Introduction: Definition of a Multivariate Random Variable551551171819420423426429430432447451454503
XUContents21.2 Discrete Multivariate RV's21.3 Mixed Multivariate Random Variables21.4 Continuous Multivariate RV's21.5 Independence21.6 Indirect Assessment of Joint Distributions21.7 Expectations21.8 Mean and Variance of a Vector RV21.9 Linear Transformations: Characteristics of a VarianceMatrixExercises55356056056356857057122The Multivariate Normal Distribution22.1 The Unit Spherical Multivariate Normal Distribution22.2 The General Nonsingular Multivariate Normal Distribution22.3 Marginal and Conditional Distributions22.4 Linear Transformations and Singular Distributions22.5 Some Comments
INTRODUCTION TO STATISTICAL DECISION THEORY John W. Pratt, Howard Raiffa, and Robert Schlaifer The MIT Press Cambridge, Massachusetts London, England . 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
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