This book has used the traditional, often referred to as frequentist, approach to statistical inference. That approach regards parameter values as fixed effects rather than random variables. Probability statements apply to possible values for the data, given the parameter values. Increasingly popular is the Bayesian alternative, which applies probability distributions to parameters as well as to data. This yields inferences in the form of probability statements about the parameters, given the data. For example, after observing the data in a clinical trial, a researcher might evaluate and report the probability that the population mean of the response variable is higher for the active drug than for the placebo.
In this chapter we first review the Bayesian approach to statistical inference. Section 10.2 presents a Bayesian analog of the normal linear model, and Section 10.3 presents Bayesian generalized linear models (GLMs). In each section we show how to obtain essentially the same substantive results as with a frequentist approach, but with Bayesian interpretations that make probability statements about the parameters. The final section presents empirical Bayes and hierarchical Bayes approaches, which make weaker assumptions about prior probability distributions for the parameters.
10.1 THE BAYESIAN APPROACH TO STATISTICAL INFERENCE
Let be a generic symbol for the parameters in a particular model, such as the effects ...