POSTERIOR MODE ESTIMATION FOR THE GENERALIZED LINEAR MODEL

D. M. EAVES^{1} AND T. CHANG^{2}

1Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

2Department of Mathematics, University of Virginia, Charlottesville, VA 22903, U.S.A.

(Received August 21, 1989; revised June 3, 1991)

Abstract. Posterior mode estimators are proposed, which arise from simply expressed prior opinion about expected outcomes, roughly as follows: a conjugate family of prior distributions is determined by a given variance function. Using a conjugate prior, a posterior mode estimator and its estimated (co-)variances are obtained through conventional maximum likelihood computations, by means of small alterations to the observed outcomes and/or to the modelled variance function. Within the conjugate family, for purposes of inference about the regression vector, a reference prior is proposed for a given choice of linear design of the canonical link. The resulting approximate reference inferences approximate the Bayesian inferences which arise from a “minimally informative” reference prior. A set of subjective prior upper and lower percentage points for the expected outcomes can be used to determine a conjugate family member. Alternatively, a set of subjective prior means and standard deviations determines a member. The subfamily of priors determinable by percentage points either includes or approximates the proposed reference prior.

Key words and phrases: Conjugate prior, contingency tables, exponential family, frequency counts, generalized linear model, Jeffreys prior, logistic regression, multinomial outcome, minimally informative prior, nonlinear regression, quasi-likelihood, reference prior, regression, variance function.