Bayesian Statistical Methods

Brian J. Reich, Sujit K. Ghosh

• Publisher: Chapman and Hall/CRC
• Year: 2019
• eBook ISBN: 9780429514340
• Topic: Matematica
• Subtopic: Matematica applicata

Bayesian Statistical Methods provides data scientists with the foundational and computational tools needed to carry out a Bayesian analysis. This book focuses on Bayesian methods applied routinely in practice including multiple linear regression, mixed effects models and generalized linear models (GLM). The authors include many examples with complete R code and comparisons with analogous frequentist procedures.

In addition to the basic concepts of Bayesian inferential methods, the book covers many general topics:

• Advice on selecting prior distributions
• Computational methods including Markov chain Monte Carlo (MCMC)
• Model-comparison and goodness-of-fit measures, including sensitivity to priors
• Frequentist properties of Bayesian methods

Case studies covering advanced topics illustrate the flexibility of the Bayesian approach:

• Semiparametric regression
• Handling of missing data using predictive distributions
• Priors for high-dimensional regression models
• Computational techniques for large datasets
• Spatial data analysis

The advanced topics are presented with sufficient conceptual depth that the reader will be able to carry out such analysis and argue the relative merits of Bayesian and classical methods. A repository of R code, motivating data sets, and complete data analyses are available on the book's website.

Brian J. Reich, Associate Professor of Statistics at North Carolina State University, is currently the editor-in-chief of the Journal of Agricultural, Biological, and Environmental Statistics and was awarded the LeRoy & Elva Martin Teaching Award.

Sujit K. Ghosh, Professor of Statistics at North Carolina State University, has over 22 years of research and teaching experience in conducting Bayesian analyses, received the Cavell Brownie mentoring award, and served as the Deputy Director at the Statistical and Applied Mathematical Sciences Institute.

1

Basics of Bayesian inference

CONTENTS

1.1 Probability background

1.1.1 Univariate distributions

1.1.1.1 Discrete distributions

1.1.1.2 Continuous distributions

1.1.2 Multivariate distributions

1.1.3 Marginal and conditional distributions

1.2 Bayes’ rule

1.2.1 Discrete example of Bayes’ rule

1.2.2 Continuous example of Bayes’rule

1.3 Introduction to Bayesian inference

1.4 Summarizing the posterior

1.4.1 Point estimation

1.4.2 Univariate posteriors

1.4.3 Multivariate posteriors

1.5 The posterior predictive distribution

1.6 Exercises

1.1 Probability background

Understanding basic probability theory is essential to any study of statistics. Generally speaking, the field of probability assumes a mathematical model for the process of interest and uses the model to compute the probability of events (e.g., what is the probability of flipping five straight heads using a fair coin?). In contrast, the field of statistics uses data to refine the probability model and test hypotheses related to the underlying process that generated the data (e.g., given we observe five straight heads, can we conclude the coin is biased?). Therefore, probability theory is a key ingredient to a statistical analysis, and in this section we review the most relevant concepts of probability for a Bayesian analysis.

Before developing probability mathematically, we briefly discuss probability from a conceptual perspective. The objective is to compute the probability of an event,$\mathcal{A},$ denoted Prob $(\mathcal{A})$. For example, we may be interested in the probability that the random variable X (random variables are generally represented with capital letters) takes the specific value x (lower-case letter), denoted Prob(X = x), or the probability that X will fall in the interval [a, b], denoted Prob(X ∈ [a, b]). There are two leading interpretations of this statement: objective and subjective. An objective interpretation views Prob $(\mathcal{A})$ as a purely mathematical statement. A frequentist interpretation is that if we repeated the experiment many times and recorded the sample proportion of the times $\mathcal{A}$ occurred, this proportion would eventually converge to the number Prob $(\mathcal{A})\in [0,1]$ as the number of samples increases. A subjective interpretation is that Prob $(\mathcal{A})$ represents an individual’s degree of belief, which is often quantified in terms of the amount the individual would be willing to wager that $\mathcal{A}$ will occur. As we will see, these two conceptual interpretations of probability parallel the two primary statistical frameworks: frequentist and Bayesian. However, a Bayesian analysis makes use of both of these concepts.

1.1.1 Univariate distribu...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1 Basics of Bayesian inference
• 2 From prior information to posterior inference
• 3 Computational approaches
• 4 Linear models
• 5 Model selection and diagnostics
• 6 Case studies using hierarchical modeling
• 7 Statistical properties of Bayesian methods
• Appendices
• Bibliography
• Index