Mathematical Theory of Bayesian Statistics

This book introduces the mathematical foundation of Bayesian statistics. It is well known that Bayesian inference is more accurate than the maximum likelihood method in many real-world problems: however, its mathematical foundations have been left unexplained. Recently, new research on Bayesian statistics uncovered the mathematical laws by which the behavior of Bayesian inference can be estimated and the advances of Bayes estimation have been clarified. This book introduces such mathematical foundations to students and researchers.

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.

• 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.

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.