1st Edition

Probability and Bayesian Modeling

By Jim Albert, Jingchen Hu Copyright 2020
    552 Pages
    by Chapman & Hall

    Continue Shopping

    Probability and Bayesian Modeling is an introduction to probability and Bayesian thinking for undergraduate students with a calculus background. The first part of the book provides a broad view of probability including foundations, conditional probability, discrete and continuous distributions, and joint distributions. Statistical inference is presented completely from a Bayesian perspective. The text introduces inference and prediction for a single proportion and a single mean from Normal sampling. After fundamentals of Markov Chain Monte Carlo algorithms are introduced, Bayesian inference is described for hierarchical and regression models including logistic regression. The book presents several case studies motivated by some historical Bayesian studies and the authors’ research.

    This text reflects modern Bayesian statistical practice. Simulation is introduced in all the probability chapters and extensively used in the Bayesian material to simulate from the posterior and predictive distributions. One chapter describes the basic tenets of Metropolis and Gibbs sampling algorithms; however several chapters introduce the fundamentals of Bayesian inference for conjugate priors to deepen understanding. Strategies for constructing prior distributions are described in situations when one has substantial prior information and for cases where one has weak prior knowledge. One chapter introduces hierarchical Bayesian modeling as a practical way of combining data from different groups. There is an extensive discussion of Bayesian regression models including the construction of informative priors, inference about functions of the parameters of interest, prediction, and model selection.

    The text uses JAGS (Just Another Gibbs Sampler) as a general-purpose computational method for simulating from posterior distributions for a variety of Bayesian models. An R package ProbBayes is available containing all of the book datasets and special functions for illustrating concepts from the book.

    A complete solutions manual is available for instructors who adopt the book in the Additional Resources section.

    1. Probability: A Measurement of Uncertainty
    2. 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


    3. Counting Methods
    4. Introduction: Rolling Dice, Yahtzee, and Roulette

      Equally Likely Outcomes

      The Multiplication Counting Rule



      Arrangements of Non-Distinct Objects

      Playing Yahtzee


    5. Conditional Probability
    6. 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


    7. Discrete Distributions
    8. 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


    9. Continuous Distributions
    10. 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


    11. Joint Probability Distributions
    12. 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


    13. Learning About a Binomial Probability
    14. 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


    15. Modeling Measurement and Count Data
    16. Introduction

      Modeling Measurements


      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


      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


      The Poisson distribution

      Bayesian inferences

      Case study: Learning about website counts


    17. Simulation by Markov Chain Monte Carlo
    18. Introduction

      The Bayesian computation problem

      Choosing a prior

      The two-parameter Normal problem

      Overview of the chapter

      Markov Chains


      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


      Graphs and summaries

      Using JAGS

      Normal sampling model

      Multiple chains

      Posterior predictive checking

      Comparing two proportions


    19. Bayesian Hierarchical Modeling
    20. 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


    21. Simple Linear Regression
    22. 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


      Prior distributions

      Posterior Analysis

      A Conditional Means Prior


    23. Bayesian Multiple Regression and Logistic Models
    24. Introduction

      Bayesian Multiple Linear Regression

      Example: expenditures of US households

      A multiple linear regression model

      Weakly informative priors and inference through MCMC


      Comparing Regression Models

      Bayesian Logistic Regression

      Example: US women labor participation

      A logistic regression model

      Conditional means priors and inference through MCMC



    25. Case Studies


              Federalist Papers Study


              Data on word use

              Poisson density sampling

              Negative Binomial sampling

              Comparison of rates for two authors

             Which words distinguish the two authors?

              Career Trajectories


              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



           Appendix A: The constant in the Beta posterior

           Appendix B: The posterior predictive distribution

           Appendix C: Comparing Bayesian models


    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."

    "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