Posterior distribution in GLMMs with non-Gaussian responses

Determine the posterior density q(γ | y) of the random effects γ given the response y in generalized linear mixed models with canonical link and a multivariate normal prior on γ when y follows a non-Gaussian exponential family distribution. Specifically, derive an exact characterization or computable expression for q(γ | y) defined by q(γ | y) = f(y | γ) π(γ) / ∫ f(y | γ) π(γ) dγ over γ ∈ ℝ^r, which remains unresolved due to the intractable integral in the denominator.

Background

Generalized linear mixed models (GLMMs) are specified by an exponential family conditional distribution for the response y with a canonical link and a multivariate normal prior for the random effects γ. For non-Gaussian responses, the posterior density q(γ | y) involves an intractable normalization integral, making analytical or exact computation difficult.

The paper introduces a Special Integral Computation (SIC) method that provides exact posterior mean and covariance for γ without computing the posterior density itself. Despite this advance, obtaining the full posterior distribution q(γ | y) remains unresolved, limiting fully Bayesian inferential procedures that require the complete posterior (e.g., credible intervals and posterior predictive checks).