Subjective and Objective Bayesian Statistics

Shorter, more concise chapters provide flexible coverage of the subject. Expanded coverage includes: uncertainty and randomness, prior distributions, predictivism, estimation, analysis of variance, and classification and imaging. Includes topics not covered in other books, such as the de Finetti Transform. Author S. James Press is the modern guru of Bayesian statistics.

S. JAMES PRESS, PhD, is a Distinguished Professor in the Department of Statistics at the University of California, Riverside. He is the author (with Judith M. Tanur) of The Subjectivity of Scientists and the Bayesian Approach, also published by John Wiley & Sons, Inc., 2001.

Preface.

Preface to the First Edition.

A Bayesian Hall of Fame.

PART I: FOUNDATIONS AND PRINCIPLES.

1. Background.

1.1 Rationale for Bayesian Inference and Preliminary Views of BayesÆ Theorem.

1.2 Example: Observing a Desired Experimental Effect.

1.3 Thomas Bayes.

1.4 Brief Descriptions of the Chapters.

2. A Bayesian Perspective on Probability.

2.1 Introduction.

2.2 Types of Probability.

2.3 Coherence.

2.4 Operationalizing Subjective Probability Beliefs.

2.5 Calibration of Probability Assessors.

2.6 Comparing Probability Definitions.

3. The Likelihood Function.

3.1 Introduction.

3.2 Likelihood Function.

3.3 Likelihood Principle.

3.4 Likelihood Principle and Conditioning.

3.5 Likelihood and Bayesian Inference.

3.6 Development of the Likelihood Function Using Histograms and Other Graphical Methods.

4. BayesÆ Theorem.

4.1 Introduction.

4.2 General Form of BayesÆ Theorem for Events.

4.3 BayesÆ Theorem for Discrete Data and Discrete Parameter.

4.4 BayesÆ Theorem for Continuous Data and Discrete Parameter.

4.5 BayesÆ Theorem for Discrete Data and Continuous Parameter.

4.6 BayesÆ Theorem for Continuous Data and Continuous Parameter.

5. Prior Distributions.

5.1 Introduction.

5.2 Objective and Subjective Prior Distributions.

5.3 (Univariate) Prior Distributions for a Single Parameter.

5.4 Prior Distributions for Vector and Matrix Parameters.

5.5 Data-Mining Priors.

5.6 Wrong Priors.

PART II: NUMERICAL IMPLEMENTATION OF THE BAYESIAN PARADIGM.

6. Markov Chain Monte Carlo Methods (Siddhartha Chib).

6.1 Introduction.

6.2 MetropolisûHastings (MûH) Algorithm.

6.3 Multiple-Block MûH Algorithm.

6.4 Some Techniques Useful in MCMC Sampling.

6.5 Examples.

6.6 Comparing Models Using MCMC Methods.

7. Large Sample Posterior Distributions and Approximations.

7.1 Introduction.

7.2 Large-Sample Posterior Distributions.

7.3 Approximate Evaluation of Bayesian Integrals.

7.4 Importance Sampling.

PART III: BAYESIAN STATISTICAL INFERENCE AND DECISION MAKING.

8. Bayesian Estimation.

8.1 Introduction.

8.2 Univariate (Point) Bayesian Estimation.

8.3 Multivariate (Point) Bayesian Estimation.

8.4 Interval Estimation.

8.5 Empirical BayesÆ Estimation.

8.6 Robustness in Bayesian Estimation.

9. Bayesian Hypothesis Testing.

9.1 Introduction.

9.2 A Brief History of Scientific Hypothesis Testing.

9.3 Problems with Frequentist Methods of Hypothesis Testing.

9.4 LindleyÆs Vague Prior Procedure for Bayesian Hypothesis Testing.

9.5 JeffreysÆ Procedure for Bayesian Hypothesis Testing.

10. Predictivism.

10.1 Introduction.

10.2 Philosophy of Predictivism.

10.3 Predictive Distributions=Comparing Theories.

10.4 Exchangeability.

10.5 De FinettiÆs Theorem.

10.6 The De Finetti Transform.

10.7 Predictive Distributions in Classification and Spatial and Temporal Analysis.

10.8 Bayesian Neural Nets.

11. Bayesian Decision Making.

11.1 Introduction.

11.2 Loss Functions.

11.3 Admissibility.

PART IV: MODELS AND APPLICATIONS.

12. Bayesian Inference in the General Linear Model.

12.1 Introduction.

12.2 Simple Linear Regression.

12.3 Multivariate Regression Model.

12.4 Multivariate Analysis of Variance Model.

12.5 Bayesian Inference in the Multivariate Mixed Model.

13. Model Averaging (Merlise Clyde).

13.1 Introduction.

13.2 Model Averaging and Subset Selection in Linear Regression.

13.3 Prior Distributions.

13.4 Posterior Distributions.

13.5 Choice of Hyperparameters.

13.6 Implementing BMA.

13.7 Examples.

14. Hierarchical Bayesian Modeling (Alan Zaslavsky).

14.1 Introduction.

14.2 Fundamental Concepts and Nomenclature.

14.3 Applications and Examples.

14.4 Inference in Hierarchical Models.

14.5 Relationship to Non-Bayesian Approaches.

14.6 Computation for Hierarchical Models.

14.7 Software for Hierarchical Models.

15. Bayesian Factor Analysis.

15.1 Introduction.

15.2 Background.

15.3 Bayesian Factor Analysis Model for Fixed Number of Factors.

15.4 Choosing the Number of Factors.

15.5 Additional Model Considerations.

16. Bayesian Inference in Classification and Discrimination.

16.1 Introduction.

16.2 Likelihood Function.

16.3 Prior Density.

16.4 Posterior Density.

16.5 Predictive Density.

16.6 Posterior Classification Probability.

16.7 Example: Two Populations.

16.8 Second Guessing Undecided Respondents: An Application.

16.9 Extensions of the Basic Classification Problem.

Description of Appendices.

Appendix 1. Bayes, Thomas, (Hilary L. Seal).

Appendix 2. Thomas Bayes. A Bibliographical Note (George A. Barnard).

Appendix 3. Communication of BayesÆ Essay to the Philosophical Transactions of the Royal Society of London (Richard Price).

Appendix 4. An Essay Towards Solving a Problem in the Doctrine of Chances (Reverend Thomas Bayes).

Appendix 5. Applications of Bayesian Statistical Science.

Appendix 6. Selecting the Bayesian Hall of Fame.

Appendix 7. Solutions to Selected Exercises.

Bibliography.

Subject Index.

Author Index.

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Subjective and Objective Bayesian Statistics Principles, Models, and Applications

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Author Biography

Table of Contents

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