Bayesian Theory

This highly acclaimed text, now available in paperback, provides a thorough account of key concepts and theoretical results, with particular emphasis on viewing statistical inference as a special case of decision theory. Information-theoretic concepts play a central role in the development of the theory, which provides, in particular, a detailed discussion of the problem of specification of so-called prior ignorance . The work is written from the authors s committed Bayesian perspective, but an overview of non-Bayesian theories is also provided, and each chapter contains a wide-ranging critical re-examination of controversial issues. The level of mathematics used is such that most material is accessible to readers with knowledge of advanced calculus. In particular, no knowledge of abstract measure theory is assumed, and the emphasis throughout is on statistical concepts rather than rigorous mathematics. The book will be an ideal source for all students and researchers in statistics, mathematics, decision analysis, economic and business studies, and all branches of science and engineering, who wish to further their understanding of Bayesian statistics

Mathematical Theory of Bayesian Statistics introduces the mathematical foundation of Bayesian inference which is well-known to be more accurate in many real-world problems than the maximum likelihood method. Recent research has uncovered several mathematical laws in Bayesian statistics, by which both the generalization loss and the marginal likelihood are estimated even if the posterior distribution cannot be approximated by any normal distribution. Features Explains Bayesian inference not subjectively but objectively. Provides a mathematical framework for conventional Bayesian theorems. Introduces and proves new theorems. Cross validation and information criteria of Bayesian statistics are studied from the mathematical point of view. Illustrates applications to several statistical problems, for example, model selection, hyperparameter optimization, and hypothesis tests. This book provides basic introductions for students, researchers, and users of Bayesian statistics, as well as applied mathematicians. Author Sumio Watanabe is a professor of Department of Mathematical and Computing Science at Tokyo Institute of Technology. He studies the relationship between algebraic geometry and mathematical statistics.

In this new edition the author has added substantial material on Bayesian analysis, including lengthy new sections on such important topics as empirical and hierarchical Bayes analysis, Bayesian calculation, Bayesian communication, and group decision making. With these changes, the book can be used as a self-contained introduction to Bayesian analysis. In addition, much of the decision-theoretic portion of the text was updated, including new sections covering such modern topics as minimax multivariate (Stein) estimation.

This book is based on lectures given at Yale in 1971-1981 to students prepared with a course in measure-theoretic probability. It contains one technical innovation-probability distributions in which the total probability is infinite. Such improper distributions arise embarras singly frequently in Bayes theory, especially in establishing correspondences between Bayesian and Fisherian techniques. Infinite probabilities create interesting complications in defining conditional probability and limit concepts. The main results are theoretical, probabilistic conclusions derived from probabilistic assumptions. A useful theory requires rules for constructing and interpreting probabilities. Probabilities are computed from similarities, using a formalization of the idea that the future will probably be like the past. Probabilities are objectively derived from similarities, but similarities are sUbjective judgments of individuals. Of course the theorems remain true in any interpretation of probability that satisfies the formal axioms. My colleague David Potlard helped a lot, especially with Chapter 13. Dan Barry read proof. vii Contents CHAPTER 1 Theories of Probability 1. 0. Introduction 1 1. 1. Logical Theories: Laplace 1 1. 2. Logical Theories: Keynes and Jeffreys 2 1. 3. Empirical Theories: Von Mises 3 1. 4. Empirical Theories: Kolmogorov 5 1. 5. Empirical Theories: Falsifiable Models 5 1. 6. Subjective Theories: De Finetti 6 7 1. 7. Subjective Theories: Good 8 1. 8. All the Probabilities 10 1. 9. Infinite Axioms 11 1. 10. Probability and Similarity 1. 11. References 13 CHAPTER 2 Axioms 14 2. 0. Notation 14 2. 1. Probability Axioms 14 2. 2.

This volume guides the reader along a statistical journey that begins with the basic structure of Bayesian theory, and then provides details on most of the past and present advances in this field.

Bayesian methods are growing more and more popular, finding new practical applications in the fields of health sciences, engineering, environmental sciences, business and economics and social sciences, among others. This book explores the use of Bayesian analysis in the statistical estimation of the unknown phenomenon of interest. The contents demonstrate that where such methods are applicable, they offer the best possible estimate of the unknown. Beyond presenting Bayesian theory and methods of analysis, the text is illustrated with a variety of applications to real world problems.

Now in its third edition, this classic book is widely considered the leading text on Bayesian methods, lauded for its accessible, practical approach to analyzing data and solving research problems. Bayesian Data Analysis, Third Edition continues to take an applied approach to analysis using up-to-date Bayesian methods. The authors—all leaders in the statistics community—introduce basic concepts from a data-analytic perspective before presenting advanced methods. Throughout the text, numerous worked examples drawn from real applications and research emphasize the use of Bayesian inference in practice. New to the Third Edition Four new chapters on nonparametric modeling Coverage of weakly informative priors and boundary-avoiding priors Updated discussion of cross-validation and predictive information criteria Improved convergence monitoring and effective sample size calculations for iterative simulation Presentations of Hamiltonian Monte Carlo, variational Bayes, and expectation propagation New and revised software code The book can be used in three different ways. For undergraduate students, it introduces Bayesian inference starting from first principles. For graduate students, the text presents effective current approaches to Bayesian modeling and computation in statistics and related fields. For researchers, it provides an assortment of Bayesian methods in applied statistics. Additional materials, including data sets used in the examples, solutions to selected exercises, and software instructions, are available on the book’s web page.

This introduction to the theory of variational Bayesian learning summarizes recent developments and suggests practical applications.