Contents
1Probability: A Measurement of Uncertainty
1.2The Classical View of a Probability
1.3The Frequency View of a Probability
1.4The Subjective View of a Probability
1.7Events and Event Operations
1.8The Three Probability Axioms
1.9The Complement and Addition Properties
2.1Introduction: Rolling Dice, Yahtzee, and Roulette
2.3The Multiplication Counting Rule
2.6Arrangements of Non-Distinct Objects
3.1Introduction: The Three Card Problem
3.4Definition and the Multiplication Rule
3.5The Multiplication Rule under Independence
3.7R Example: Learning about a Spinner
4.1Introduction: The Hat Check Problem
4.2Random Variable and Probability Distribution
4.3Summarizing a Probability Distribution
4.4Standard Deviation of a Probability Distribution
4.5.3Mean and standard deviation of a binomial
4.5.4Negative binomial experiments
5.1Introduction: A Baseball Spinner Game
5.3Probability Density: Waiting for a Bus
5.4The Cumulative Distribution Function
5.5Summarizing a Continuous Random Variable
5.7Binomial Probabilities and the Normal Curve
5.8Sampling Distribution of the Mean
6Joint Probability Distributions
6.2Joint Probability Mass Function: Sampling from a Box
6.5Independence and Measuring Association
6.6Flipping a Random Coin: The Beta-Binomial Distribution
6.7Bivariate Normal Distribution
7Learning about a Binomial Probability
7.1Introduction: Thinking Subjectively about a Proportion
7.2Bayesian Inference with Discrete Priors
7.2.1Example: students’ dining preference
7.2.2Discrete prior distributions for proportion p
7.2.3Likelihood of proportion p
7.2.4Posterior distribution for proportion p
7.2.5Inference: students’ dining preference
7.2.6Discussion: using a discrete prior
7.3.1The beta distribution and probabilities
7.3.2Choosing a beta density to represent prior opinion
7.4.2From beta prior to beta posterior: conjugate priors
7.5Bayesian Inferences with Continuous Priors
7.5.1Bayesian hypothesis testing
7.5.2Bayesian credible intervals
8Modeling Measurement and Count Data
8.3Bayesian Inference with Discrete Priors
8.3.1Example: Roger Federer’s time-to-serve
8.3.2Simplification of the likelihood
8.3.3Inference: Federer’s time-to-serve
8.4.1The normal prior for mean μ
8.5.2A quick peak at the update procedure
8.6Bayesian Inferences for Continuous Normal Mean
8.6.1Bayesian hypothesis testing and credible interval
8.7Posterior Predictive Checking
8.8.4Case study: Learning about website counts
9Simulation by Markov Chain Monte Carlo
9.1.1The Bayesian computation problem
9.1.3The two-parameter normal problem
9.2.3Simulating a Markov chain
9.3.1Example: Walking on a number line
9.3.3A general function for the Metropolis algorithm
9.4Example: Cauchy-Normal Problem
9.4.1Choice of starting value and proposal region
9.4.2Collecting the simulated draws
9.5.1Bivariate discrete distribution
9.5.3Normal sampling - both parameters unknown
9.6MCMC Inputs and Diagnostics
9.6.1Burn-in, starting values, and multiple chains
9.7.3Posterior predictive checking
9.7.4Comparing two proportions
10Bayesian Hierarchical Modeling
10.1.2Example: standardized test scores
10.1.5A two-stage prior leading to compromise estimates
10.2Hierarchical Normal Modeling
10.2.1Example: ratings of animation movies
10.2.2A hierarchical Normal model with random σ
10.3Hierarchical Beta-Binomial Modeling
10.3.1Example: Deaths after heart attacks
10.3.2A hierarchical beta-binomial model
11.2Example: Prices and Areas of House Sales
11.3A Simple Linear Regression Model
11.4A Weakly Informative Prior
11.7Bayesian Inferences with Simple Linear Regression
11.7.1Simulate fits from the regression model
11.7.2Learning about the expected response
11.7.3Prediction of future response
11.7.4Posterior predictive model checking
12Bayesian Multiple Regression and Logistic Models
12.2Bayesian Multiple Linear Regression
12.2.1Example: expenditures of U.S. households
12.2.2A multiple linear regression model
12.2.3Weakly informative priors and inference through MCMC
12.3Comparing Regression Models
12.4Bayesian Logistic Regression
12.4.1Example: U.S. women labor participation
12.4.2A logistic regression model
12.4.3Conditional means priors and inference through MCMC
13.2.3Poisson density sampling
13.2.4Negative binomial sampling
13.2.5Comparison of rates for two authors
13.2.6Which words distinguish the two authors?
13.3.2Measuring hitting performance in baseball
13.3.3A hitter’s career trajectory
13.3.4Estimating a single trajectory
13.3.5Estimating many trajectories by a hierarchical model
13.4.1Two classes of test takers
13.4.2A latent class model with two classes
13.4.3Disputed authorship of the Federalist Papers
14.1Appendix A: The constant in the beta posterior
14.2Appendix B: The posterior predictive distribution