Statistica Sinica 13(2003), 461-476

CONJUGATE PRIORS FOR GENERALIZED LINEAR MODELS

Ming-Hui Chen and Joseph G. Ibrahim

University of Connecticut and University of North Carolina

Abstract: We propose a novel class of conjugate priors for the family of generalized linear models. Properties of the priors are investigated in detail and elicitation issues are examined. We establish theorems characterizing the propriety and existence of moments of the priors under various settings, examine asymptotic properties of the priors, and investigate the relationship to normal priors. Our approach is based on the notion of specifying a prior prediction $y_{0}$ for the response vector of the current study, and a scalar precision parameter $a_{0}$ which quantifies one's prior belief in $y_0$. Then $(y_{0},a_{0})$, along with the covariate matrix $X$ of the current study, are used to specify the conjugate prior for the regression coefficients $\beta$ in a generalized linear model. We examine properties of the prior for $a_0$ fixed and for $a_0$ random, and study elicitation strategies for $(y_0, a_0)$ in detail. We also study generalized linear models with an unknown dispersion parameter. An example is given to demonstrate the properties of the prior and the resulting posterior.

Key words and phrases: Conjugate prior, generalized linear models, Gibbs sampling, historical data, logistic regression, poisson regression, predictive elicitation.