Introduction to Bayesian Statistics

Rating:
Summary: I learned a great deal about how the Bayesians do statistics
Review: Approximately ten years ago, I received my initial statistics instruction from Dr. Robert Hogg, one of the leading educators in the field. There were occasions in class when he referred to the Bayesians, calling them a group of statisticians who rely on separate "a priori" and "a posteriori" analyses. As was his style, he made several jokes about "a posteriori" data. The structure of the class was such that he could not spend a great deal of time on Bayesian statistics, but his brief comments have always remained in my mind.
Therefore, when I received this book I immediately decided that I would read it. From it, I learned that the Bayesian approach to statistics is valuable and more accurately reflects the way humans think about the world. There are two primary philosophical approaches to statistics, the frequentist and Bayesian, with the frequentist being that most widely covered in basic statistics classes. A frequentist statistician uses random samples to provide estimates for unknown parameters of populations.
The Bayesian approach considers the population parameters to be random variables. The process of determining the value of a parameter starts with a subjective prior distribution of the parameter before the data is analyzed. After the data is collected and organized, Bayes' theorem is then used to revise your beliefs about the values of the parameters.
The first sections deal with the basics of summarizing and displaying data; logic, probability and uncertainty. These sections are generally not different from frequentist statistics, so there is no distinction between the Bayesian and frequentist philosophies. The first real differences occur at the end of chapter 5, which covers logic, probability and uncertainty. This is the point where Bayes' theorem is introduced and the principles of prior and posterior probabilities. Chapter 5 describes discrete random variables, and again, this section is standard material on probability.
The true philosophy of Bayesian statistics appears in chapter 6, which covers Bayesian inference for discrete random variables. As a newcomer to this area, I read it with great interest and learned a great deal about how Bayesian operations are performed. The remaining sections deal with the processes of performing basic statistical operations using Bayesian methods. This includes:

* Bayesian inference for binomial proportion.
* Bayesian inference for normal mean.
* Bayesian inference for difference between means.
* Bayesian inference for simple linear regression.

There are also two chapters that compare the Bayesian and frequentist techniques. Chapter 9 compares the Bayesian and fequentist techniques for the inference for proportions and chapter 11 compares the techniques for the inference for means. Exercises are included at the end of each chapter and appendix F is devoted to the answers to odd-numbered exercises.
I learned an enormous amount about Bayesian methods from this book and I strongly recommend it if you are interested in learned how the Bayesians do things.

Published in the recreational mathematics e-mail newsletter, reprinted with permission.