Introduction to Bayesian Statistics, 3rd Edition

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"...this edition is useful and effective in teaching Bayesian inference at both elementary and intermediate levels. It is a well-written book on elementary Bayesian inference, and the material is easily accessible. It is both concise and timely, and provides a good collection of overviews and reviews of important tools used in Bayesian statistical methods."

There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. In this third Edition, four newly-added chapters address topics that reflect the rapid advances in the field of Bayesian statistics. The authors continue to provide a Bayesian treatment of introductory statistical topics, such as scientific data gathering, discrete random variables, robust Bayesian methods, and Bayesian approaches to inference for discrete random variables, binomial proportions, Poisson, and normal means, and simple linear regression. In addition, more advanced topics in the field are presented in four new chapters: Bayesian inference for a normal with unknown mean and variance; Bayesian inference for a Multivariate Normal mean vector; Bayesian inference for the Multiple Linear Regression Model; and Computational Bayesian Statistics including Markov Chain Monte Carlo. The inclusion of these topics will facilitate readers' ability to advance from a minimal understanding of Statistics to the ability to tackle topics in more applied, advanced level books. Minitab macros and R functions are available on the book's related website to assist with chapter exercises. Introduction to Bayesian Statistics, Third Edition also features:

  • Topics including the Joint Likelihood function and inference using independent Jeffreys priors and join conjugate prior
  • The cutting-edge topic of computational Bayesian Statistics in a new chapter, with a unique focus on Markov Chain Monte Carlo methods
  • Exercises throughout the book that have been updated to reflect new applications and the latest software applications
  • Detailed appendices that guide readers through the use of R and Minitab software for Bayesian analysis and Monte Carlo simulations, with all related macros available on the book's website

Introduction to Bayesian Statistics, Third Edition is a textbook for upper-undergraduate or first-year graduate level courses on introductory statistics course with a Bayesian emphasis. It can also be used as a reference work for statisticians who require a working knowledge of Bayesian statistics.

Preface xiii

1 Introduction to Statistical Science 1

1.1 The Scientic Method: A Process for Learning 3

1.2 The Role of Statistics in the Scientic Method 5

1.3 Main Approaches to Statistics 5

1.4 Purpose and Organization of This Text 8

2 Scientic Data Gathering 13

2.1 Sampling from a Real Population 14

2.2 Observational Studies and Designed Experiments 17

Monte Carlo Exercises 23

3 Displaying and Summarizing Data 31

3.1 Graphically Displaying a Single Variable 32

3.2 Graphically Comparing Two Samples 39

3.3 Measures of Location 41

3.4 Measures of Spread 44

3.5 Displaying Relationships Between Two or More Variables 46

3.6 Measures of Association for Two or More Variables 49

Exercises 52

4 Logic, Probability, and Uncertainty 59

4.1 Deductive Logic and Plausible Reasoning 60

4.2 Probability 62

4.3 Axioms of Probability 64

4.4 Joint Probability and Independent Events 65

4.5 Conditional Probability 66

4.6 Bayes' Theorem 68

4.7 Assigning Probabilities 74

4.8 Odds and Bayes Factor 75

4.9 Beat the Dealer 76

Exercises 80

5 Discrete Random Variables 83

5.1 Discrete Random Variables 84

5.2 Probability Distribution of a Discrete Random Variable 86

5.3 Binomial Distribution 90

5.4 Hypergeometric Distribution 92

5.5 Poisson Distribution 93

5.6 Joint Random Variables 96

5.7 Conditional Probability for Joint Random Variables 100

Exercises 104

6 Bayesian Inference for Discrete Random Variables 109

6.1 Two Equivalent Ways of Using Bayes' Theorem 114

6.2 Bayes' Theorem for Binomial with Discrete Prior 116

6.3 Important Consequences of Bayes' Theorem 119

6.4 Bayes' Theorem for Poisson with Discrete Prior 120

Exercises 122

Computer Exercises 126

7 Continuous Random Variables 129

7.1 Probability Density Function 131

7.2 Some Continuous Distributions 135

7.3 Joint Continuous Random Variables 143

7.4 Joint Continuous and Discrete Random Variables 144

Exercises 147

8 Bayesian Inference for Binomial Proportion 149

8.1 Using a Uniform Prior 150

8.2 Using a Beta Prior 151

8.3 Choosing Your Prior 154

8.4 Summarizing the Posterior Distribution 158

8.5 Estimating the Proportion 161

8.6 Bayesian Credible Interval 162

Exercises 164

Computer Exercises 167

9 Comparing Bayesian and Frequentist Inferences for Proportion 169

9.1 Frequentist Interpretation of Probability and Parameters 170

9.2 Point Estimation 171

9.3 Comparing Estimators for Proportion 174

9.4 Interval Estimation 175

9.5 Hypothesis Testing 178

9.6 Testing a One-Sided Hypothesis 179

9.7 Testing a Two-Sided Hypothesis 182

Exercises 187

Monte Carlo Exercises 190

10 Bayesian Inference for Poisson 193

10.1 Some Prior Distributions for Poisson 194

10.2 Inference for Poisson Parameter 200

Exercises 207

Computer Exercises 208

11 Bayesian Inference for Normal Mean 211

11.1 Bayes' Theorem for Normal Mean with a Discrete Prior 211

11.2 Bayes' Theorem for Normal Mean with a Continuous Prior 218

11.3 Choosing Your Normal Prior 222

11.4 Bayesian Credible Interval for Normal Mean 224

11.5 Predictive Density for Next Observation 227

Exercises 230

Computer Exercises 232

12 Comparing Bayesian and Frequentist Inferences for Mean 237

12.1 Comparing Frequentist and Bayesian Point Estimators 238

12.2 Comparing Condence and Credible Intervals for Mean 241

12.3 Testing a One-Sided Hypothesis about a Normal Mean 243

12.4 Testing a Two-Sided Hypothesis about a Normal Mean 247

Exercises 251

13 Bayesian Inference for Di erence Between Means 255

13.1 Independent Random Samples from Two Normal Distributions 256

13.2 Case 1: Equal Variances 257

13.3 Case 2: Unequal Variances 262

13.4 Bayesian Inference for Dierence Between Two Proportions Using Normal Approximation 265

13.5 Normal Random Samples from Paired Experiments 266

Exercises 272

14 Bayesian Inference for Simple Linear Regression 283

14.1 Least Squares Regression 284

14.2 Exponential Growth Model 288

14.3 Simple Linear Regression Assumptions 290

14.4 Bayes' Theorem for the Regression Model 292

14.5 Predictive Distribution for Future Observation 298

Exercises 303

Computer Exercises 312

15 Bayesian Inference for Standard Deviation 315

15.1 Bayes' Theorem for Normal Variance with a Continuous Prior 316

15.2 Some Specic Prior Distributions and the Resulting Posteriors 318

15.3 Bayesian Inference for Normal Standard Deviation 326

Exercises 332

Computer Exercises 335

16 Robust Bayesian Methods 337

16.1 Eect of Misspecied Prior 338

16.2 Bayes' Theorem with Mixture Priors 340

Exercises 349

Computer Exercises 351

17 Bayesian Inference for Normal with Unknown Mean and Variance 355

17.1 The Joint Likelihood Function 358

17.2 Finding the Posterior when Independent Jeffreys' Priors for μ and σ2 Are Used 359

17.3 Finding the Posterior when a Joint Conjugate Prior for μ and σ2 Is Used 361

17.4 Difference Between Normal Means with Equal Unknown Variance 367

17.5 Difference Between Normal Means with Unequal Unknown Variances 377

Computer Exercises 383

Appendix: Proof that the Exact Marginal Posterior Distribution of μ is Student's t 385

18 Bayesian Inference for Multivariate Normal Mean Vector 393

18.1 Bivariate Normal Density 394

18.2 Multivariate Normal Distribution 397

18.3 The Posterior Distribution of the Multivariate Normal Mean Vector when Covariance Matrix Is Known 398

18.4 Credible Region for Multivariate Normal Mean Vector when Covariance Matrix Is Known 400

18.5 Multivariate Normal Distribution with Unknown Covariance Matrix 402

Computer Exercises 406

19 Bayesian Inference for the Multiple Linear Regression Model 411

19.1 Least Squares Regression for Multiple Linear Regression Model 412

19.2 Assumptions of Normal Multiple Linear Regression Model 414

19.3 Bayes' Theorem for Normal Multiple Linear Regression Model 415

19.4 Inference in the Multivariate Normal Linear Regression Model 419

19.5 The Predictive Distribution for a Future Observation 425

Computer Exercises 428

20 Computational Bayesian Statistics Including Markov Chain Monte Carlo 431

20.1 Direct Methods for Sampling from the Posterior 436

20.2 Sampling - Importance - Resampling 450

20.3 Markov Chain Monte Carlo Methods 454

20.4 Slice Sampling 470

20.5 Inference from a Posterior Random Sample 473

20.6 Where to Next? 475

A Introduction to Calculus 477

B Use of Statistical Tables 497

C Using the Included Minitab Macros 523

D Using the Included R Functions 543

E Answers to Selected Exercises 565

References 591

Index 595

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