Introduction to Bayesian Statistics

Praise for the First Edition "I cannot think of a better book for teachers of introductory statistics who want a readable and pedagogically sound text to introduce Bayesian statistics." —Statistics in Medical Research "[This book] is written in a lucid conversational style, which is so rare in mathematical writings. It does an excellent job of presenting Bayesian statistics as a perfectly reasonable approach to elementary problems in statistics." —STATS: The Magazine for Students of Statistics, American Statistical Association "Bolstad offers clear explanations of every concept and method making the book accessible and valuable to undergraduate and graduate students alike." —Journal of Applied Statistics The use of Bayesian methods in applied statistical analysis has become increasingly popular, yet most introductory statistics texts continue to only present the subject using frequentist methods. Introduction to Bayesian Statistics, Second Edition focuses on Bayesian methods that can be used for inference, and it also addresses how these methods compare favorably with frequentist alternatives. Teaching statistics from the Bayesian perspective allows for direct probability statements about parameters, and this approach is now more relevant than ever due to computer programs that allow practitioners to work on problems that contain many parameters. This book uniquely covers the topics typically found in an introductory statistics book—but from a Bayesian perspective—giving readers an advantage as they enter fields where statistics is used. This Second Edition provides: Extended coverage of Poisson and Gamma distributions Two new chapters on Bayesian inference for Poisson observations and Bayesian inference for the standard deviation for normal observations A twenty-five percent increase in exercises with selected answers at the end of the book A calculus refresher appendix and a summary on the use of statistical tables New computer exercises that use R functions and Minitab® macros for Bayesian analysis and Monte Carlo simulations Introduction to Bayesian Statistics, Second Edition is an invaluable textbook for advanced undergraduate and graduate-level statistics courses as well as a practical reference for statisticians who require a working knowledge of Bayesian statistics.

Table of Contents:
Preface. Preface to First Edition. 1. Introduction to Statistical Science. 1.1 The Scientific Method: A Process for Learning. 1.2 The Role of Statistics in the Scientific Method. 1.3 Main Approaches to Statistics. 1.4 Purpose and Organization of This Text. 2. Scientific Data Gathering. 2.1 Sampling from a Real Population. 2.2 Observational Studies and Designed Experiments. Monte Carlo Exercises. 3. Displaying and Summarizing Data. 3.1 Graphically Displaying a Single Variable. 3.2 Graphically Comparing Two Samples. 3.3 Measures of Location. 3.4 Measures of Spread. 3.5 Displaying Relationships Between Two or More Variables. 3.6 Measures of Association for Two or More Variables. Exercises. 4. Logic, Probability, and Uncertainty. 4.1 Deductive Logic and Plausible Reasoning. 4.2 Probability. 4.3 Axioms of Probability. 4.4 Joint Probability and Independent Events. 4.5 Conditional Probability. 4.6 Bayes' Theorem. 4.7 Assigning Probabilities. 4.8 Odds Ratios and Bayes Factor. 4.9 Beat the Dealer. Exercises. 5. Discrete Random Variables. 5.1 Discrete Random Variables. 5.2 Probability Distribution of a Discrete Random Variable. 5.3 Binomial Distribution. 5.4 Hypergeometric Distribution. 5.5 Poisson Distribution. 5.6 Joint Random Variables. 5.7 Conditional Probability for Joint Random Variables. Exercises. 6. Bayesian Inference for Discrete Random Variables. 6.1 Two Equivalent Ways of Using Bayes' Theorem. 6.2 Bayes' Theorem for Binomial with Discrete Prior. 6.3 Important Consequences of Bayes' Theorem. 6.4 Bayes' theorem for Poisson with Discrete Prior. Exercises. Computer Exercises. 7. Continuous Random Variables. 7.1 Probability Density Function. 7.2 Some Continuous Distributions. 7.3 Joint Continuous Random Variables. 7.4 Joint Continuous and Discrete Random Variables. Exercises. 8. Bayesian Inference for Binomial Proportion. 8.1 Using a Uniform Prior. 8.2 Using a Beta Prior. 8.3 Choosing Your Prior. 8.4 Summarizing the Posterior Distribution. 8.5 Estimating the Proportion. 8.6 Bayesian Credible Interval. Exercises. Computer Exercises. 9. Comparing Bayesian and Frequentist Inferences for Proportion. 9.1 Frequentist Interpretation of Probability and Parameters. 9.2 Point Estimation. 9.3 Comparing Estimators for Proportion. 9.4 Interval Estimation. 9.5 Hypothesis Testing. 9.6 Testing a OneSided Hypothesis. 9.7 Testing a TwoSided Hypothesis. Exercises. Monte Carlo Exercises. 10. Bayesian Inference for Poisson. 10.1 Some Prior Distributions for Poisson. 10.2 Inference for Poisson Parameter. Exercises. Computer Exercises. 11. Bayesian Inference for Normal Mean. 11.1 Bayes' Theorem for Normal Mean with a Discrete Prior. 11.2 Bayes' Theorem for Normal Mean with a Continuous Prior. 11.3 Choosing Your Normal Prior. 11.4 Bayesian Credible Interval for Normal Mean. 11.5 Predictive Density for Next Observation. Exercises. Computer Exercises. 12. Comparing Bayesian and Frequentist Inferences for Mean. 12.1 Comparing Frequentist and Bayesian Point Estimators. 12.2 Comparing Confidence and Credible Intervals for Mean. 12.3 Testing a OneSided Hypothesis about a Normal Mean. 12.4 Testing a TwoSided Hypothesis about a Normal Mean. Exercises. 13. Bayesian Inference for Difference between Means. 13.1 Independent Random Samples from Two Normal Distributions. 13.2 Case 1: Equal Variances. 13.3 Case 2: Unequal Variances. 13.4 Bayesian Inference for Difference Between Two Proportions Using Normal Approximation. 13.5 Normal Random Samples from Paired Experiments. Exercises. 14. Bayesian Inference for Simple Linear Regression. 14.1 Least Squares Regression. 14.2 Exponential Growth Model. 14.3 Simple Linear Regression Assumptions. 14.4 Bayes' Theorem for the Regression Model. 14.5 Predictive Distribution for Future Observation. Exercises. Computer Exercises. 15. Bayesian Inference for Standard Deviation. 15.1 Bayes' Theorem for Normal Variance with a Continuous Prior. 15.2 Some Specific Prior Distributions and The Resulting Posteriors. 15.3 Bayesian inference for normal standard deviation. Exercises. Computer Exercises. 16. Robust Bayesian Methods. 16.1 Effect of Misspecified Prior. 16.2 Bayes’ Theorem with Mixture Priors. Exercises. Computer Exercises. A. Introduction to Calculus. B. Use of Statistical Tables. C. Using the Included Minitab Macros. D. Using the Included R Functions. E. Answers to Selected Exercises. References. Topic Index.