Probability and Bayesian Modeling

Probability: A Measurement of Uncertainty
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

Counting Methods
Introduction: Rolling Dice, Yahtzee, and Roulette

Equally Likely Outcomes

The Multiplication Counting Rule

Permutations

Combinations

Arrangements of Non-Distinct Objects

Playing Yahtzee

Exercises

Conditional Probability
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

Discrete Distributions
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

Continuous Distributions
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

Joint Probability Distributions
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

Learning About a Binomial Probability
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

Modeling Measurement and Count Data
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

Simulation by Markov Chain Monte Carlo
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

Bayesian Hierarchical Modeling
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

Simple Linear Regression
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

Bayesian Multiple Regression and Logistic Models
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

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