Marginal (likelihood) distribution of the response in GLMMs

Evaluate the marginal distribution of the response y, defined as f̄(y) = ∫ f(y | γ) π(γ) dγ over γ ∈ ℝ^r, in generalized linear mixed models with canonical link and a multivariate normal prior on γ when y follows a non-Gaussian exponential family distribution. Establish an exact method to compute this likelihood function, which remains unresolved due to the intractable integral.

Background

In GLMMs with non-Gaussian responses, the marginal distribution f̄(y) (the likelihood) requires integrating out the random effects γ under a normal prior. This integral is typically intractable, obstructing exact maximum likelihood estimation and complicating Bayesian computation.

The work presented circumvents the need for f̄(y) by directly computing exact posterior moments of γ via optimization, but it does not solve the marginal likelihood integral itself. As a result, the fundamental task of exact likelihood evaluation for such GLMMs remains an open problem.