On Bayesian Analysis of Generalized Linear Models Using Jeffreys's Prior

Abstract Generalized linear models (GLM's) have proved suitable for modeling various kinds of data consisting of exponential family response variables with covariates. Bayesian analysis of such data requires specification of a prior for the regression parameters in the model used. Uniform priors are very commonly used as conventional noninformative priors. We show, however, that uniform priors for GLM's can lead to improper posterior distributions thus making them undesirable. Alternative reference priors may be constructed from Jeffreys's rule. In this article, we give two theorems that support the use of Jeffreys's priors for GLM's with intrinsically fixed or known scale parameters. These theorems provide (i) sufficient and (ii) necessary and sufficient conditions for the propriety of the (i) posterior and (ii) prior distributions as well as for the existence of moments. Implications of these theorems for some commonly used GLM's are discussed. Finally, an illustrative example is given for the binomial model with canonical link (i.e., logistic regression), and results using Jeffreys's priors are compared with those based on other non-informative and informative priors.