- Probability: A Measurement of Uncertainty
- Counting Methods
- Conditional Probability
- Discrete Distributions
- Continuous Distributions
- Joint Probability Distributions
- Learning About a Binomial Probability
- Modeling Measurement and Count Data
- Simulation by Markov Chain Monte Carlo
- Bayesian Hierarchical Modeling
- Simple Linear Regression
- Bayesian Multiple Regression and Logistic Models
- Case Studies
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
Introduction: Rolling Dice, Yahtzee, and Roulette
Equally Likely Outcomes
The Multiplication Counting Rule
Permutations
Combinations
Arrangements of Non-Distinct Objects
Playing Yahtzee
Exercises
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
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
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
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
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
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
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
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
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
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
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
Biography
Jim Albert is a Distinguished University Professor of Statistics at Bowling Green State University. His research interests include Bayesian modeling and applications of statistical thinking in sports. He has authored or coauthored several books including Ordinal Data Modeling, Bayesian Computation with R, and Workshop Statistics: Discovery with Data, A Bayesian Approach.
Jingchen (Monika) Hu is an Assistant Professor of Mathematics and Statistics at Vassar College. She teaches an undergraduate-level Bayesian Statistics course at Vassar, which is shared online across several liberal arts colleges. Her research focuses on dealing with data privacy issues by releasing synthetic data.
"The book can be used by upper undergraduate and graduate students as well as researchers and practitioners in statistics and data science from all disciplines…A background of calculus is required for the reader but no experience in programming is needed. The writing style of the book is extremely reader friendly. It provides numerous illustrative examples, valuable resources, a rich collection of materials, and a memorable learning experience."
~Technometrics"Over many years, I have wondered about the following: Should a first undergraduate course in statistics be a Bayesian course? After reading this book, I have come to the conclusion that the answer is…yes!... this is very well written textbook that can also be used as self-learning material for practitioners. It presents a clear, accessible, and entertaining account of the interplay of probability, computations, and statistical inference from the Bayesian perspective."
~ISCB News
