INSEAD Library

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Applied Statistical Decision Theory Contents Foreword Preface and Introduction Part I: Experimentation and Decision: General Theory 1. The Problem and the Two Basic Modes of Analysis 1. Description of the Decision Problem 1: The basic data; 2: Assessment of probability measures; 3: Example; 4: The general decision problem as a game. 2. Analysis in Extensive Form 1: Backwards induction; 2: Example. 3. Analysis in Normal Form 1: Decision rules; 2: Performance, error, and utility characteristics; 3: Example; 4: Equivalence of the extensive and normal form; 5: Bayesian decision theory as a completion of classical theory; 6: Informal choice of a decision rule. 4. Combination of Formal and Informal Analysis 1: Unknown costs; cutting the decision tree; 2: Incomplete analysis of the decision tree; 3: Example. 5. Prior Weights and Consistent Behavior 2. Sufficient Statistics and Noninformative Stopping 1. Introduction 1: Simplifying assumptions; 2: Bayes' theorem; kernels 2. Sufficiency 1: Bayesian definition of sufficiency; 2: Identification of sufficient statistics; 3: Equivalence of the Bayesian and classical definitions of sufficiency; 4: Nuisance parameters and marginal sufficiency. 3. Noninformative Stopping 1: Data-generating processes and stopping processes; 2: Likelihood of a sample; 3: Noninformative stopping processes; 4: Contrast between the Bayesian and classical treatments of stopping; 5: Summary. 3. Conjugate Prior Distributions 1. Introduction; Assumptions and Definitions 1: Desiderata for a family of prior distributions; 2: Sufficient statistics of fixed dimensionality. 28 3 vi 43 Contents 2. Conjugate Prior Distributions 1: Use of the sample kernel as a prior kernel; 2: The posterior distribution when the prior distribution is natural-conjugate; 3: Extension of the domain of the parameter; 4: Extension by introduction of a new parameter; 5: Conspectus of natural-conjugate densities. 3. Choice and Interpretation of a Prior Distribution 1: Distributions fitted to historical relative frequencies; 2: Distributions fitted to subjective betting odds; 3: Comparison of the weights of prior and sample evidence; 4: "Quantity of information" and "vague" opinions; 5: Sensitivity analysis; 6: Scientific reporting. 4. Analysis in Extensive Form when the Prior Distribution and Sample Likelihood are Conjugate 1: Definitions of terminal and preposterior analysis; 2: Terminal analysis; 3: Preposterior analysis. Part II: Extensive-Form Analysis When Sampling and Terminal Utilities Are Additive 4. Additive Utility, Opportunity Loss, and the Value of Information: Introduction to Part II 1. Basic Assumptions 2. Applicability of Additive Utilities 3. Computation of Expected Utility 4. Opportunity Loss 1: Definition of opportunity loss; 2: Extensive-form analysis using opportunity loss instead of utility; 3: Opportunity loss when terminal and sampling utilities are additive; 4: Direct assessment of terminal opportunity losses; 5: Upper bounds on optimal sample size. 5. The Value of Information 1: The value of perfect information; 2: The value of sample information and the net gain of sampling; 3: Summary of relations among utilities, opportunity losses, and value of information. 5A. Linear Terminal Analysis 1. Introduction 1: The transformed state description ; 2: Terminal analysis. 2. Expected Value of Perfect Information when is Scalar 1: Two-action problems; 2: Finite-action problems; 3: Evaluation of linearloss integrals; 4: Examples. 3. Preposterior Analysis 1: The posterior mean as a random variable; 2: The expected value of sample information. 4. The Prior Distribution of the Posterior Mean for Given e 1: Mean and variance of w; 2: Limiting behavior of the distribution; 3: Limiting behavior of integrals when is scalar; 4: Exact distributions of w; 5: Approximations to the distribution of w; 6: Examples. 93 79 Contents 5. Optimal Sample Size in Two-Action Problems when the Sample Observations are Normal and Their Variance is Known 1: Definitions and notation; 2: Behavior of net gain as a function of sample size; 3: Optimal sample size; 4: Asymptotic behavior of optimal sample size; 5: Asymptotic behavior of opportunity loss; 6: Fixed element in sampling cost. 6. Optimal Sample Size in Two-Action Problems when the Sample Observations are Binomial 1: Definitions and notation; 2: Behavior of the EVSI as a function of n; 3: Behavior of the net gain of sampling; optimal sample size; 4: A normal approximation to optimal sample size. 5B. Selection of the Best of Several Processes 139 7. Introduction; Basic Assumptions 8. Analysis in Terms of Differential Utility 1: Notation: the random variables v and 2: Analysis in terms of v and v; 3: The usefulness of differential utility. 9. Distribution of S and v" when the Processes are Independent Normal and is Linear in v 1: Basic assumptions; notation; 2: Conjugate distribution of v; 3: Distribution of ; 4: Distribution of g" when all processes are to be sampled; 5: Distribution of g" when some processes are not to be sampled. 10. Value of Information and Optimal Size when There are Two Independent Normal Processes 1: EVPI; 2: EVSI; 3: Optimal allocation of a fixed experimental budget; 4: Optimal sample size when h is known and only one process is to be sampled; 5: Optimal sample size when h is known and both processes are to be sampled according to l+ or l-; 6: The general problem of optimal sample size when h is known. 11. Value of Information when There are Three Independent-Normal Processes 1: The basic integral in the nondegenerate case; 2: Transformation to a unitspherical distribution; 3: Evaluation of the EVI by numerical integration; 4: Evaluation of the EVI by bivariate Normal tables when h is known; 5: Bounds on the EVI; 6: Example; 7: EVI when the prior expected utilities are equal; 8: EVSI when only one process is to be sampled. 12. Value of Information when There are More than Three Independent-Normal Processes 1: The nondegenerate case; 2: The degenerate case; 3: Choice of the optimal experiment. 6. Problems in Which the Act and State Spaces Coincide 176 1. Introduction 1: Basic assumptions; 2: Example. 2. Certainty Equivalents and Point Estimation 1: Certainty equivalents; 2: Example; 3: General theory of certainty equivalents; 4: Approximation of Xi; 5: Subjective evaluation of Xi; 6: Rough and ready estimation; 7: Multipurpose estimation. 3. Quadratic Terminal Opportunity Loss 1: Terminal analysis; 2: Preposterior analysis; 3: Optimal sample size. Contents 4. Linear Terminal Opportunity Loss 1: Terminal analysis; 2: Preposterior analysis; 3: Optimal sample size. 5. Modified Linear and Quadratic Loss Structures Part III: Distribution Theory 7. Univariate Normalized Mass and Density Functions 211 0. Introduction 1: Normalized mass and density functions; 2: Cumulative functions; 3: Moments; 4: Expectations and variances; 5: Integrand transformations; 6: Effect of linear transformations on moments. A. Natural Univariate Mass and Density Functions 213 1. Binomial Function 2. Pascal Function 3. Beta Functions 1: Standardized beta function; 2: Beta function in alternate notation. 4. Inverted Beta Functions 1: Inverted-beta-1 function; 2: Inverted-beta-2 function; 3: F function. 5. Poisson Function 6. Gamma Functions 1: Standardized gamma function; 2: Gamma-1 function; 3: Chi-square function; 4: Gamma-2 function. 7. Inverted Gamma Functions 1: Inverted-gamma-1 function; 2: Inverted-gamma-2 function. 8. Normal Functions 1: Standardized Normal functions; 2: General Normal functions. B. Compound Univariate Mass and Density Functions 9. Student Functions 1: Standardized Student function; 2: General Student function. 10. Negative-Binomial Function 11. Beta-Binomial and Beta-Pascal Functions 1: Relations with the hypergeometric function; 2: Computation of the cumulative beta-binomial and beta-Pascal functions. 8. Multivariate Normalized Density Functions 0. Introduction 1: Matrix and vector notation; 2: Inverses of matrices; 3: Positive-definite and positive-semidefinite matrices; 4: Projections; 5: Notation for multivariate densities and integrals; 6: Moments; expectations and variances. 1. Unit-Spherical Normal Function 1: Conditional and marginal densities; 2: Tables. 2. General Normal Function 1: Conditional and marginal densities; 2: Tables; 3: Linear combinations of normal random variables. 242 232 Contents 3. Student Function 1: Conditional and marginal densities; 2: Linear combinations of Student random variables. 4. Inverted-Student Function 9. Bernoulli Process 1. Prior and Posterior Analysis 1: Definition of a Bernoulli process; 2: Likelihood of a sample; 3: Conjugate distribution of p;4: Conjugate distribution of 1/p = p 2. Sampling Distributions and Preposterior Analysis: Binomial Sampling 1: Definition of binomial sampling; 2: Conditional distribution of (tip); 3: Unconditional distribution of r; 4: Distribution of p"; 5: Distribution of p". 3. Sampling Distributions and Preposterior Analysis: Pascal Sampling 1: Definition of Pascal sampling; 2: Conditional distribution of (/p); 3: Unconditional distribution of ; 4: Distribution of p", 5: Distribution of p". 10. Poisson Process 1. Prior and Posterior Analysis 1: Definition of a Poisson process; 2: Likelihood of sample; 3: Conjugate distribution of X; 4: Conjugate distribution of 1/ = 2. Sampling Distributions and Preposterior Analysis: Gamma Sampling 1: Definition of Gamma sampling; 2: Conditional distribution of (/); 3: Unconditional distribution of t; 4: Distribution of X"; 5: Distribution of 3. Sampling Distributions and Preposterior Analysis: Poisson Sampling 1: Definition of Poisson sampling; 2: Conditional distribution of (/); 3: Unconditional distribution of f; 4: Distribution of ."; 5: Distribution of /". 11. Independent Normal Process A. Mean Known 1. Prior and Posterior Analysis 1: Definition of an independent Normal process; 2: Likelihood of a sample when is known; 3: Conjugate distribution of h; 4: Conjugate distribution of . 2. Sampling Distributions and Preposterior Analysis with Fixed v 1: Conditional distribution of (/h); 2: Unconditional distribution of ; 3: Distribution of v". B. Precision Known 3. Prior and Posterior Analysis 1: Likelihood of a sample when h is known; 2: Conjugate distribution of 4. Sampling Distributions and Preposterior Analysis with Fixed n 1: Conditional distribution of (m/); 2: Unconditional distribution of m; 3: Distribution of m and ". 294 290 290 275 261 Contents C. Neither Parameter Known 5. Prior and Posterior Analysis 1: Likelihood of a sample when neither parameter is known; 2: Likelihood of the incomplete statistics (m, n) and (v, v); 3: Distribution of (A, h); 4: Marginal distribution of h; 5: Marginal distribution of 1.i; 6: Limiting behavior of the prior distribution. 6. Sampling Distributions with Fixed n 1: Conditional joint distribution of (A, 151/4, h); 2: Unconditional joint distribution of (A, 10; 3: Unconditional distribution of A and 0. 7. Preposterior Analysis with Fixed n 1: Joint distribution of (A", a"); 2: Distribution of fit" and 6"; 3: Distribution of a"; 4: Distribution of ft". 12. Independent Multinormal Process 310 298 A. Precision Known 310 1. Prior and Posterior Analysis 1: Definition of the independent multinormal process; 2: Likelihood of a sample when fl is known; 3: Likelihood of a sample when both h and are known; 4: Conjugate distribution of 2. Sampling Distributions with Fixed n 1: Conditional distribution of (Alp); 2: Unconditional distribution of hi. 3. Preposterior Analysis with Fixed n 1: Distribution of rii" ; 2: Distribution of ft". B. Relative Precision Known 4. Prior and Posterior Analysis 1: Likelihood of a sample when only is known; 2: Likelihood of the statistics (m, n) and (v, v); 3: Conjugate distribution of (a, h); 4: Distributions of Ii; 5: Distributions of ,1. 5. Sampling Distributions with Fixed n 1: Conditional joint distribution of (th, h); 2: Unconditional joint distribution of (th, ii); 3: Unconditional distributions of m" and 0. 6. Preposterior Analysis with Fixed n 1: Joint distribution of (m", v); 2: Distributions of m" and 0"; 3: Distributions of u." and u". C. Interrelated Univariate Normal Processes 7. Introduction 8. Analysis When All Processes Are Sampled 9. Analysis when Only p < r Processes are Sampled 1: Notation; 2: Posterior analysis; 3: Conditional sampling distributions with fixed n; 4: Marginal distribution of (filth) and (pi, A); 5: Unconditional distributions of /hi and 0; 6: Distributions of thin and 0"; 7: Preposterior analysis. 326 316 Contents 13. Normal Regression Process 1. Introduction 1: Definition of the normal regression process; 2: Likelihood of a sample; 3: Analogy with the multinormal process. A. Precision Known 2. Prior and Posterior Analysis 1: Likelihood of a sample when h is known; 2: Distribution of ß. 3. Sampling Distributions with Fixed X 1: Conditional distribution of (y/ß); 2: Unconditional distribution of y; 3: Distributions of b when X is of rank r. 4. Preposterior Analysis with Fixed X of Rank r 1: Distribution of b"; 2: Distribution of ß". 336 334 B. Precision Unknown 342 5. Prior and Posterior Analysis 1: Likelihood of a sample when neither ß nor h is known; 2: Distribution of (ß, h); 3: Marginal and conditional distributions of h; 4: Marginal and con ditional distributions of ß. 6. Sampling Distributions with Fixed X 1: Unconditional distribution of y; 2: Conditional joint distribution of (b, v/ß, h); 3: Unconditional distributions of b and v. 7. Preposterior Analysis with Fixed X of Rank r 1: Joint distribution of (b", v") ; 2: Distributions of b" and v"; 3: Distributions of ß" and ß". C. Xt' X Singular 349 8. Introduction 1: Definitions and Notation. 9. Distributions of b* and v 1: Conditional distributions of b* and 2: Distributions of (ß*/h) and (ß*, h); 3: Unconditional distributions of b* and v. 10. Preposterior Analysis 1: Utilities dependent on b" alone; 2: Utilities dependent on (b", v") jointly; 3: Distribution of