Theory of Gaussian variational approximation for a Poisson mixed model

Likelihood-based inference for the parameters of generalized linear mixed models is hindered by the presence of intractable integrals. Gaussian variational approximation provides a fast and effective means of approximate inference. We provide some theory for this type of approximation for a simple Poisson mixed model. In particular, we establish consistency at rate m−1/2 + n−1, where m is the number of groups and n is the number of repeated measurements.