Probability and Bayesian Modeling

2. Probability: A Measurement of Uncertainty

3. Introduction

   The Classical View of a Probability

   The Frequency View of a Probability

   The Subjective View of a Probability

   The Sample Space

   Assigning Probabilities

   Events and Event Operations

   The Three Probability Axioms

   The Complement and Addition Properties

   Exercises

4. Counting Methods

5. Introduction: Rolling Dice, Yahtzee, and Roulette

   Equally Likely Outcomes

   The Multiplication Counting Rule

   Permutations

   Combinations

   Arrangements of Non-Distinct Objects

   Playing Yahtzee

   Exercises

6. Conditional Probability

7. Introduction: The Three Card Problem

   In Everyday Life

   In a Two-Way Table

   Definition and the Multiplication Rule

   The Multiplication Rule Under Independence

   Learning Using Bayes' Rule

   R Example: Learning About a Spinner

   Exercises

8. Discrete Distributions

9. Introduction: The Hat Check Problem

   Random Variable and Probability Distribution

   Summarizing a Probability Distribution

   Standard Deviation of a Probability Distribution

   Coin-Tossing Distributions

   Binomial probabilities

   Binomial computations

   Mean and standard deviation of a Binomial

   Negative Binomial Experiments

   Exercises

10. Continuous Distributions

11. Introduction: A Baseball Spinner Game

    The Uniform Distribution

    Probability Density: Waiting for a Bus

    The Cumulative Distribution Function

    Summarizing a Continuous Random Variable

    Normal Distribution

    Binomial Probabilities and the Normal Curve

    Sampling Distribution of the Mean

    Exercises

12. Joint Probability Distributions

13. Introduction

    Joint Probability Mass Function: Sampling From a Box

    Multinomial Experiments

    Joint Density Functions

    Independence and Measuring Association

    Flipping a Random Coin: The Beta-Binomial Distribution

    Bivariate Normal Distribution

    Exercises

14. Learning About a Binomial Probability

15. Introduction: Thinking About a Proportion Subjectively

    Bayesian Inference with Discrete Priors

    Example: students' dining preference

    Discrete prior distributions for proportion p

    Likelihood of proportion p

    Posterior distribution for proportion p

    Inference: students' dining preference

    Discussion: using a discrete prior

    Continuous Priors

    The Beta distribution and probabilities

    Choosing a Beta density curve to represent prior opinion

    Updating the Beta Prior

    Bayes' rule calculation

    From Beta prior to Beta posterior: conjugate priors

    Bayesian Inferences with Continuous Priors

    Bayesian hypothesis testing

    Bayesian credible intervals

    Bayesian prediction

    Predictive Checking

    Exercises

16. Modeling Measurement and Count Data

17. Introduction

    Modeling Measurements

    Examples

    The general approach

    Outline of chapter

    Bayesian Inference with Discrete Priors

    Example: Roger Federer's time-to-serve

    Simplification of the likelihood

    Inference: Federer's time-to-serve

    Continuous Priors

    The Normal prior for mean _

    Choosing a Normal prior

    Updating the Normal Prior

    Introduction

    A quick peak at the update procedure

    Bayes' rule calculation

    Conjugate Normal prior

    Bayesian Inferences for Continuous Normal Mean

    Bayesian hypothesis testing and credible interval

    Bayesian prediction

    Posterior Predictive Checking

    Modeling Count Data

    Examples

    The Poisson distribution

    Bayesian inferences

    Case study: Learning about website counts

    Exercises

18. Simulation by Markov Chain Monte Carlo

19. Introduction

    The Bayesian computation problem

    Choosing a prior

    The two-parameter Normal problem

    Overview of the chapter

    Markov Chains

    Definition

    Some properties

    Simulating a Markov chain

    The Metropolis Algorithm

    Example: Walking on a number line

    The general algorithm

    A general function for the Metropolis algorithm

    Example: Cauchy-Normal problem

    Choice of starting value and proposal region

    Collecting the simulated draws

    Gibbs Sampling

    Bivariate discrete distribution

    Beta-Binomial sampling

    Normal sampling { both parameters unknown

    MCMC Inputs and Diagnostics

    Burn-in, starting values, and multiple chains

    Diagnostics

    Graphs and summaries

    Using JAGS

    Normal sampling model

    Multiple chains

    Posterior predictive checking

    Comparing two proportions

    Exercises

20. Bayesian Hierarchical Modeling

21. Introduction

    Observations in groups

    Example: standardized test scores

    Separate estimates?

    Combined estimates?

    A two-stage prior leading to compromise estimates

    Hierarchical Normal Modeling

    Example: ratings of animation movies

    A hierarchical Normal model with random _

    Inference through MCMC

    Hierarchical Beta-Binomial Modeling

    Example: Deaths after heart attack

    A hierarchical Beta-Binomial model

    Inference through MCMC

    Exercises

22. Simple Linear Regression

23. Introduction

    Example: Prices and Areas of House Sales

    A Simple Linear Regression Model

    A Weakly Informative Prior

    Posterior Analysis

    Inference through MCMC

    Bayesian Inferences with Simple Linear Regression

    Simulate fits from the regression model

    Learning about the expected response

    Prediction of future response

    Posterior predictive model checking

    Informative Prior

    Standardization

    Prior distributions

    Posterior Analysis

    A Conditional Means Prior

    Exercises

24. Bayesian Multiple Regression and Logistic Models

25. Introduction

    Bayesian Multiple Linear Regression

    Example: expenditures of US households

    A multiple linear regression model

    Weakly informative priors and inference through MCMC

    Prediction

    Comparing Regression Models

    Bayesian Logistic Regression

    Example: US women labor participation

    A logistic regression model

    Conditional means priors and inference through MCMC

    Prediction

    Exercises

26. Case Studies

Introduction

Federalist Papers Study

Introduction

Data on word use

Poisson density sampling

Negative Binomial sampling

Comparison of rates for two authors

Which words distinguish the two authors?

Career Trajectories

Introduction

Measuring hitting performance in baseball

A hitter's career trajectory

Estimating a single trajectory

Estimating many trajectories by a hierarchical model

Latent Class Modeling

Two classes of test takers

A latent class model with two classes

Disputed authorship of the Federalist Papers

Exercises

Appendix

Appendix A: The constant in the Beta posterior

Appendix B: The posterior predictive distribution

Appendix C: Comparing Bayesian models