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POSTERIOR MODE ESTIMATION FOR THE GENERALIZED

LINEAR MODEL

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

^{1} *Department of Mathematics and Statistics, Simon Fraser University,*

Burnaby, British Columbia, Canada V5A 1S6

^{2} *Department 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.

**Source**
( TeX ,
DVI ,
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