Bayes Rules!

An Introduction to Applied Bayesian Modeling

Alicia A. Johnson, Miles Q. Ott, Mine Dogucu

• Publisher: Chapman and Hall/CRC
• Year: 2022
• Print ISBN: 9781032191591
• eBook ISBN: 9781000529562
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

An engaging, sophisticated, and fun introduction to the field of Bayesian statistics, Bayes Rules!: An Introduction to Applied Bayesian Modeling brings the power of modern Bayesian thinking, modeling, and computing to a broad audience. In particular, the book is an ideal resource for advanced undergraduate statistics students and practitioners with comparable experience. the book assumes that readers are familiar with the content covered in a typical undergraduate-level introductory statistics course. Readers will also, ideally, have some experience with undergraduate-level probability, calculus, and the R statistical software. Readers without this background will still be able to follow along so long as they
are eager to pick up these tools on the fly as all R code is provided.Bayes Rules! empowers readers to weave Bayesian approaches into their everyday practice. Discussions and applications are data driven. A natural progression from fundamental to multivariable, hierarchical models emphasizes a practical and generalizable model building process. The evaluation of these Bayesian models reflects the fact that a data analysis does not exist in a vacuum.

Features

• Utilizes data-driven examples and exercises.

• Emphasizes the iterative model building and evaluation process.

• Surveys an interconnected range of multivariable regression and classification models.

• Presents fundamental Markov chain Monte Carlo simulation.

• Integrates R code, including RStan modeling tools and the bayesrules package.

• Encourages readers to tap into their intuition and learn by doing.

• Provides a friendly and inclusive introduction to technical Bayesian concepts.

• Supports Bayesian applications with foundational Bayesian theory.

Unit IIIBayesian Regression & Classification

9 Simple Normal Regression

DOI: 10.1201/9780429288340-9

Welcome to Unit 3!

Our work in Unit 1 (learning how to think like Bayesians and build simple Bayesian models) and Unit 2 (exploring how to simulate and analyze these models), sets us up to expand our Bayesian toolkit to more sophisticated models in Unit 3. Thus, far, our models have focused on the study of a single data variable Y. For example, in Chapter 4 we studied Y, whether or not films pass the Bechdel test. Yet once we have a grip on the variability in Y, we often have follow-up questions: can the passing / failing of the Bechdel test be explained by a film's budget, genre, release date, etc.?

In general, we often want to model the relationship between some response variable Y and predictors ($X_1,X_2,\mathrm{\dots },X_p$). This is the shared goal of the remaining chapters, which will survey a broad set of Bayesian modeling tools that we conventionally break down into two tasks:

• Regression tasks are those that analyze and predict quantitative response variables (e.g., Y = hippocampal volume).
• Classification tasks are those that analyze categorical response variables with the goal of predicting or classifying the response category (e.g., classify Y, whether a news article is real or fake).

We'll survey a few Bayesian regression techniques in Chapters 9 through 12: Normal, Poisson, and Negative Binomial regression. We'll also survey two Bayesian classification techniques in Chapters 13 and 14: logistic regression and naive Bayesian classification. Though we can't hope to introduce you to every regression and classification tool you'll ever need, the five we've chosen here are generalizable to a broader set of applications. At the outset of this exploration, we encourage you to focus on the Bayesian modeling principles. By focusing on principles over a perceived set of rules (which don't exist), you'll empower yourself to extend and apply what you learn here beyond the scope of this book.

In Chapter 9 we'll start with the foundational Normal regression model for a quantitative response variable Y. Consider the following data story. Capital Bikeshare is a bike sharing service in the Washington, D.C. area. To best serve its registered members, the company must understand the demand for its service. To help them out, we can analyze the number of rides taken on a random sample of n days, ($Y_1,Y_2,\mathrm{\dots },Y_n$). Since Y_i is a count variable, you might assume that ridership might be well modeled by a Poisson. However, past bike riding seasons have exhibited bell-shaped daily ridership with a variability in ridership that far exceeds the typical ridership, grossly violating the Poisson assumption of equal mean and variance (5.4). Thus, we'll assume instead that, independently from day to day, the number of rides varies normally around some typical ridership, μ, with standard deviationσ (Figure 9.1):$Y_i|\mu ,\sigma \stackrel{ind}{~}N(\mu ,{\sigma }^2).$

FIGURE 9.1: A Normal model of bike ridership.

Utilizing the Normal-Normal model from Chapter 5, we could conduct ...

Table of contents

• Cover Page
• Half-Title Page
• Series Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Foreword
• Preface
• About the Authors
• I Bayesian Foundations
• II Posterior Simulation & Analysis
• III Bayesian Regression & Classification
• IV Hierarchical Bayesian models
• Bibliography
• Index