Introduction to Bayesian Statistics
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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 s2 Are Used 359
17.3 Finding the Posterior when a Joint Conjugate Prior for µ and s2 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
