Introduction to Bayesian Statistics

There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. In Bayesian statistics the rules of probability are used to make inferences about the parameter. Prior information about the parameter and sample information from the data are combined using Bayes theorem. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. This book uniquely covers the topics usually found in a typical introductory statistics book but from a Bayesian perspective.

Contents

Preface.

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 Event s.

4.5 Conditional Probability.

4.6 Bayes’ Theorem.

4.7 Assigning Probabilities.

4.8 Odds Ratios and Bayes Factor.

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 Joint Random Variables.

5.6 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.

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 Normal Mean.

10.1 Bayes’ Theorem for Normal Mean with a Discrete Prior.

10.2 Bayes’ Theorem for Normal Mean with a Continuous Prior.

10.3 Choosing Your Normal Prior.

10.4 Bayesian Credible Interval for Normal Mean.

10.5 Predictive Density for Next Observation.

Exercises.

Computer Exercises.

11. Comparing Bayesian and Frequentist Inferences for Mean.

11.1 Comparing Frequentist and Bayesian Point Estimators.

11.2 Comparing Confidence and Credible Intervals for Mean.

11.3 Testing a OneSided Hypothesis about a Normal Mean.

11.4 Testing a TwoSided Hypothesis about a Normal Mean.

Exercises.

12. Bayesian Inference for Difference between Means.

12.1 Independent Random Samples from Two Normal Distributions.

12.2 Case 1: Equal Variances.

12.3 Case 2: Unequal Variances.

12.4 Bayesian Inference for Difference Between Two Proportions Using Normal Approximation.

12.5 Normal Random Samples from Paired Experiments.

Exercises.

13. Bayesian Inference for Simple Linear Regression.

13.1 Least Squares Regression.

13.2 Exponential Growth Model.

13.3 Simple Linear Regression Assumptions.

13.4 Bayes’ Theorem for the Regression Model.

13.5 Predictive Distribution for Future Observation.

Exercises.

14. Robust Bayesian Methods.

14.1 Effect of Misspecified Prior.

14.2 Bayes’ Theorem with Mixture Priors.

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.

Index.