Learn the science of collecting information to make effective decisions Everyday decisions are made without the benefit of accurate information. Optimal Learning develops the needed principles for gathering information to make decisions, especially when collecting information is time-consuming and expensive. Designed for readers with an elementary background in probability and statistics, the book presents effective and practical policies illustrated in a wide range of applications, from energy, homeland security, and transportation to engineering, health, and business. This book covers the fundamental dimensions of a learning problem and presents a simple method for testing and comparing policies for learning. Special attention is given to the knowledge gradient policy and its use with a wide range of belief models, including lookup table and parametric and for online and offline problems. Three sections develop ideas with increasing levels of sophistication: *Fundamentals explores fundamental topics, including adaptive learning, ranking and selection, the knowledge gradient, and bandit problems*Extensions and Applications features coverage of linear belief models, subset selection models, scalar function optimization, optimal bidding, and stopping problems *Advanced Topics explores complex methods including simulation optimization, active learning in mathematical programming, and optimal continuous measurements Each chapter identifies a specific learning problem, presents the related, practical algorithms for implementation, and concludes with numerous exercises. A related website features additional applications and downloadable software, including MATLAB and the Optimal Learning Calculator, a spreadsheet-based package that provides an introduction to learning and a variety of policies for learning.
Preface xv Acknowledgments xix 1 The challenges of learning 1 1.1 Learning the best path 2 1.2 Areas of application 4 1.3 Major problem classes 12 1.4 The different types of learning 13 1.5 Learning from different communities 16 1.6 Information collection using decision trees 18 1.6.1 A basic decision tree 18 1.6.2 Decision tree for offline learning 20 1.6.3 Decision tree for online learning 21 1.6.4 Discussion 25 1.7 Website and downloadable software 26 1.8 Goals of this book 26 Problems 28 2 Adaptive learning 31 2.1 The frequentist view 32 2.2 The Bayesian view 33 2.2.1 The updating equations for independent beliefs 34 2.2.2 The expected value of information 36 2.2.3 Updating for correlated normal priors 38 2.2.4 Bayesian updating with an uninformative prior 41 2.3 Updating for non-Gaussian priors 42 2.3.1 The gamma-exponential model 43 2.3.2 The gamma-Poisson model 44 2.3.3 The Pareto-uniform model 45 2.3.4 Models for learning probabilities 46 2.3.5 Learning an unknown variance 49 2.4 Monte Carlo simulation 51 2.5 Why does it work? 54 2.5.1 Derivation of ~- 54 2.5.2 Derivation of Bayesian updating equations for independent beliefs 55 2.6 Bibliographic notes 57 Problems 57 3 The economics of information 61 3.1 An elementary information problem 61 3.2 The marginal value of information 65 3.3 An information acquisition problem 68 3.4 Bibliographic notes 70 Problems 70 4 Ranking and selection 71 4.1 The model 72 4.2 Measurement policies 75 4.2.1 Deterministic vs. sequential policies 75 4.2.2 Optimal sequential policies 76 4.2.3 Heuristic policies 77 4.3 Evaluating policies 81 4.4 More advanced topics 83 4.4.1 An alternative representation of the probability space 83 4.4.2 Equivalence of using true means and sample estimates 84 4.5 Bibliographic notes 85 Problems 85 5 The knowledge gradient 89 5.1 The knowledge gradient for independent beliefs 90 5.1.1 Computation 91 5.1.2 Some properties of the knowledge gradient 93 5.1.3 The four distributions of learning 94 5.2 The value of information and the S-curve effect 95 5.3 Knowledge gradient for correlated beliefs 98 5.4 The knowledge gradient for some non-Gaussian distributions 103 5.4.1 The gamma-exponential model 104 5.4.2 The gamma-Poisson model 107 5.4.3 The Pareto-uniform model 108 5.4.4 The beta-Bernoulli model 109 5.4.5 Discussion 111 5.5 Relatives of the knowledge gradient 112 5.5.1 Expected improvement 113 5.5.2 Linear loss 114 5.6 Other issues 116 5.6.1 Anticipatory vs. experiential learning 117 5.6.2 The problem of priors 118 5.6.3 Discussion 120 5.7 Why does it work? 121 5.7.1 Derivation of the knowledge gradient formula 121 5.8 Bibliographic notes 125 Problems 126 6 Bandit problems 139 6.1 The theory and practice of Gittins indices 141 6.1.1 Gittins indices in the beta-Bernoulli model 142 6.1.2 Gittins indices in the normal-normal model 145 6.1.3 Approximating Gittins indices 147 6.2 Variations of bandit problems 148 6.3 Upper confidence bounding 149 6.4 The knowledge gradient for bandit problems 151 6.4.1 The basic idea 151 6.4.2 Some experimental comparisons 153 6.4.3 Non-normal models 156 6.5 Bibliographic notes 157 Problems 157 7 Elements of a learning problem 163 7.1 The states of our system 164 7.2 Types of decisions 166 7.3 Exogenous information 167 7.4 Transition functions 168 7.5 Objective functions 168 7.5.1 Designing versus controlling 168 7.5.2 Measurement costs 170 7.5.3 Objectives 170 7.6 Evaluating policies 175 7.7 Discussion 177 7.8 Bibliographic notes 178 Problems 178 8 Linear belief models 181 8.1 Applications 182 8.1.1 Maximizing ad clicks 182 8.1.2 Dynamic pricing 184 8.1.3 Housing loans 184 8.1.4 Optimizing dose response 185 8.2 A brief review of linear regression 186 8.2.1 The normal equations 186 8.2.2 Recursive least squares 187 8.2.3 A Bayesian interpretation 188 8.2.4 Generating a prior 189 8.3 The knowledge gradient for a linear model 191 8.4 Application to drug discovery 192 8.5 Application to dynamic pricing 196 8.6 Bibliographic notes 200 P
