Reference Bayesian Methods for Generalized Linear Mixed Models
Ranjini Natarajan and Robert E. Kass
Bayesian methods furnish an
attractive approach to inference in generalized linear mixed models.
In the absence of
subjective prior information for the random effect variance
components, these analyses are typically conducted using
either the standard invariant prior for normal responses
or diffuse conjugate priors.
Previous work has pointed out serious difficulties with
and we show here that, as in normal mixed models,
the standard invariant prior leads to an improper posterior
distribution for generalized linear mixed models.
The purpose of this paper is
to propose and investigate two alternate reference (i.e.,
``objective'' or ``noninformative'') priors: an approximate uniform
shrinkage prior and an approximate Jeffreys's prior.
We give conditions for the
existence of the posterior distribution under any prior for the
variance components in conjunction with
a uniform prior for the fixed effects. The
approximate uniform shrinkage prior is shown to satisfy these
conditions for several families of distributions,
in some cases
under mild constraints on the data. Simulation studies conducted using
a logit-normal model
reveal that the approximate uniform shrinkage prior
improves substantially upon a plug-in empirical Bayes rule and fully
Bayesian methods using diffuse conjugate specifications. The
methodology is illustrated on a seizure data set.
Keywords: Conjugate prior;
Hierarchical models; Jeffreys's prior;
Uniform shrinkage prior; Variance components
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