(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.