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Sparse variational approximations allow for principled and scalable inferencein Gaussian Process (GP) models. In settings where several GPs are part of thegenerative model, theses GPs are a posteriori coupled. For many applicationssuch as regression where predictive accuracy is the quantity of interest, thiscoupling is not crucial. Howewer if one is interested in posterior uncertainty,it cannot be ignored. A key element of variational inference schemes is thechoice of the approximate posterior parameterization. When the number of latentvariables is large, mean field (MF) methods provide fast and accurate posteriormeans while more structured posterior lead to inference algorithm of greatercomputational complexity. Here, we extend previous sparse GP approximations andpropose a novel parameterization of variational posteriors in the multi-GPsetting allowing for fast and scalable inference capturing posteriordependencies.

Structured Variational Inference for Coupled Gaussian Processes