Generalized Linear Models

Generalized linear models are extensions of the linear regression model described in the previous chapter. In particular, they avoid the selection of a single transformation of the data that must achieve the possibly conflicting goals of normality and linearity imposed by the linear regression model, which is for instance impossible for binary or count responses. The trick that allows both a feasible processing and an extension of linear regression is first to turn the covariates into a real number by a linear projection and then to transform this value so that it fits the support of the response. We focus here on the Bayesian analysis of probit and logit models for binary data and log-linear models for contingency tables.

On the methodological side, we present a general MCMC method, the Metropolis–Hastings algorithm, which is used for the simulation of complex distributions where both regular and Gibbs sampling fail. This includes in particular the random walk Metropolis–Hastings algorithm, which acts like a plain vanilla MCMC algorithm.