Probability and Bayesian Modeling by Albert, Jim -- Read -- Imperial Library of Trantor

Index

Cover Half Title Series Page Title Page Copyright Page Contents Preface 1. Probability: A Measurement of Uncertainty

1.1 Introduction 1.2 The Classical View of a Probability 1.3 The Frequency View of a Probability 1.4 The Subjective View of a Probability 1.5 The Sample Space 1.6 Assigning Probabilities 1.7 Events and Event Operations 1.8 The Three Probability Axioms 1.9 The Complement and Addition Properties 1.10 Exercises

2. Counting Methods

2.1 Introduction: Rolling Dice, Yahtzee, and Roulette 2.2 Equally Likely Outcomes 2.3 The Multiplication Counting Rule 2.4 Permutations 2.5 Combinations 2.6 Arrangements of Non-Distinct Objects 2.7 Playing Yahtzee 2.8 Exercises

3. Conditional Probability

3.1 Introduction: The Three Card Problem 3.2 In Everyday Life 3.3 In a Two-Way Table 3.4 Definition and the Multiplication Rule 3.5 The Multiplication Rule under Independence 3.6 Learning Using Bayes’ Rule 3.7 R Example: Learning about a Spinner 3.8 Exercises

4. Discrete Distributions

4.1 Introduction: The Hat Check Problem 4.2 Random Variable and Probability Distribution 4.3 Summarizing a Probability Distribution 4.4 Standard Deviation of a Probability Distribution 4.5 Coin-Tossing Distributions

4.5.1 Binomial probabilities 4.5.2 Binomial computations 4.5.3 Mean and standard deviation of a binomial 4.5.4 Negative binomial experiments

4.6 Exercises

5. Continuous Distributions

5.1 Introduction: A Baseball Spinner Game 5.2 The Uniform Distribution 5.3 Probability Density: Waiting for a Bus 5.4 The Cumulative Distribution Function 5.5 Summarizing a Continuous Random Variable 5.6 Normal Distribution 5.7 Binomial Probabilities and the Normal Curve 5.8 Sampling Distribution of the Mean 5.9 Exercises

6. Joint Probability Distributions

6.1 Introduction 6.2 Joint Probability Mass Function: Sampling from a Box 6.3 Multinomial Experiments 6.4 Joint Density Functions 6.5 Independence and Measuring Association 6.6 Flipping a Random Coin: The Beta-Binomial Distribution 6.7 Bivariate Normal Distribution 6.8 Exercises

7. Learning about a Binomial Probability

7.1 Introduction: Thinking Subjectively about a Proportion 7.2 Bayesian Inference with Discrete Priors

7.2.1 Example: students’ dining preference 7.2.2 Discrete prior distributions for proportion p 7.2.3 Likelihood of proportion p 7.2.4 Posterior distribution for proportion p 7.2.5 Inference: students’ dining preference 7.2.6 Discussion: using a discrete prior

7.3 Continuous Priors

7.3.1 The beta distribution and probabilities 7.3.2 Choosing a beta density to represent prior opinion

7.4 Updating the Beta Prior

7.4.1 Bayes’ rule calculation 7.4.2 From beta prior to beta posterior: conjugate priors

7.5 Bayesian Inferences with Continuous Priors

7.5.1 Bayesian hypothesis testing 7.5.2 Bayesian credible intervals 7.5.3 Bayesian prediction

7.6 Predictive Checking 7.7 Exercises

8. Modeling Measurement and Count Data

8.1 Introduction 8.2 Modeling Measurements

8.2.1 Examples 8.2.2 The general approach 8.2.3 Outline of chapter

8.3 Bayesian Inference with Discrete Priors

8.3.1 Example: Roger Federer’s time-to-serve 8.3.2 Simplification of the likelihood 8.3.3 Inference: Federer’s time-to-serve

8.4 Continuous Priors

8.4.1 The normal prior for mean μ 8.4.2 Choosing a normal prior

8.5 Updating the Normal Prior

8.5.1 Introduction 8.5.2 A quick peak at the update procedure 8.5.3 Bayes’ rule calculation 8.5.4 Conjugate normal prior

8.6 Bayesian Inferences for Continuous Normal Mean

8.6.1 Bayesian hypothesis testing and credible interval 8.6.2 Bayesian prediction

8.7 Posterior Predictive Checking 8.8 Modeling Count Data

8.8.1 Examples 8.8.2 The Poisson distribution 8.8.3 Bayesian inferences 8.8.4 Case study: Learning about website counts

8.9 Exercises

9. Simulation by Markov Chain Monte Carlo

9.1 Introduction

9.1.1 The Bayesian computation problem 9.1.2 Choosing a prior 9.1.3 The two-parameter normal problem 9.1.4 Overview of the chapter

9.2 Markov Chains

9.2.1 Definition 9.2.2 Some properties 9.2.3 Simulating a Markov chain

9.3 The Metropolis Algorithm

9.3.1 Example: Walking on a number line 9.3.2 The general algorithm 9.3.3 A general function for the Metropolis algorithm

9.4 Example: Cauchy-Normal Problem

9.4.1 Choice of starting value and proposal region 9.4.2 Collecting the simulated draws

9.5 Gibbs Sampling

9.5.1 Bivariate discrete distribution 9.5.2 Beta-binomial sampling 9.5.3 Normal sampling - both parameters unknown

9.6 MCMC Inputs and Diagnostics

9.6.1 Burn-in, starting values, and multiple chains 9.6.2 Diagnostics 9.6.3 Graphs and summaries

9.7 Using JAGS

9.7.1 Normal sampling model 9.7.2 Multiple chains 9.7.3 Posterior predictive checking 9.7.4 Comparing two proportions

9.8 Exercises

10. Bayesian Hierarchical Modeling

10.1 Introduction

10.1.1 Observations in groups 10.1.2 Example: standardized test scores 10.1.3 Separate estimates? 10.1.4 Combined estimates? 10.1.5 A two-stage prior leading to compromise estimates

10.2 Hierarchical Normal Modeling

10.2.1 Example: ratings of animation movies 10.2.2 A hierarchical Normal model with random σ 10.2.3 Inference through MCMC

10.3 Hierarchical Beta-Binomial Modeling

10.3.1 Example: Deaths after heart attacks 10.3.2 A hierarchical beta-binomial model 10.3.3 Inference through MCMC

10.4 Exercises

11. Simple Linear Regression

11.1 Introduction 11.2 Example: Prices and Areas of House Sales 11.3 A Simple Linear Regression Model 11.4 A Weakly Informative Prior 11.5 Posterior Analysis 11.6 Inference through MCMC 11.7 Bayesian Inferences with Simple Linear Regression

11.7.1 Simulate fits from the regression model 11.7.2 Learning about the expected response 11.7.3 Prediction of future response 11.7.4 Posterior predictive model checking

11.8 Informative Prior

11.8.1 Standardization 11.8.2 Prior distributions 11.8.3 Posterior Analysis

11.9 A Conditional Means Prior 11.10 Exercises

12. Bayesian Multiple Regression and Logistic Models

12.1 Introduction 12.2 Bayesian Multiple Linear Regression

12.2.1 Example: expenditures of U.S. households 12.2.2 A multiple linear regression model 12.2.3 Weakly informative priors and inference through MCMC 12.2.4 Prediction

12.3 Comparing Regression Models 12.4 Bayesian Logistic Regression

12.4.1 Example: U.S. women labor participation 12.4.2 A logistic regression model 12.4.3 Conditional means priors and inference through MCMC 12.4.4 Prediction

12.5 Exercises

13. Case Studies

13.1 Introduction 13.2 Federalist Papers Study

13.2.1 Introduction 13.2.2 Data on word use 13.2.3 Poisson density sampling 13.2.4 Negative binomial sampling 13.2.5 Comparison of rates for two authors 13.2.6 Which words distinguish the two authors?

13.3 Career Trajectories

13.3.1 Introduction 13.3.2 Measuring hitting performance in baseball 13.3.3 A hitter’s career trajectory 13.3.4 Estimating a single trajectory 13.3.5 Estimating many trajectories by a hierarchical model

13.4 Latent Class Modeling

13.4.1 Two classes of test takers 13.4.2 A latent class model with two classes 13.4.3 Disputed authorship of the Federalist Papers

13.5 Exercises

14. Appendices

14.1 Appendix A: The constant in the beta posterior 14.2 Appendix B: The posterior predictive distribution 14.3 Appendix C: Comparing Bayesian models

Bibliography Index

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