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 predictionfor the response vector of the current study, and a scalar precision parameter
which quantifies one's prior belief in
. Then
, along with the covariate matrix
of the current study, are used to specify the conjugate prior for the regression coefficients
in a generalized linear model. We examine properties of the prior for
fixed and for
random, and study elicitation strategies for
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.