Introduction to Bayesian Statistics

Content: 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    and   2 Are Used 359    17.3 Finding the Posterior when a Joint Conjugate Prior for    and   2 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    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