The Variational Gaussian Approximation Revisited.
Manfred Opper and Cedric Archambeau
The variational approximation of posterior distributions by multivariate Gaussians has been much less popular in the Machine Learning community compared to the corresponding approximation by factorising distributions. This is for a good reason: the Gaussian approximation is in general plagued by an $\Ocal(N^2)$ number of variational parameters to be optimised, $N$ being the number of random variables. In this work, we discuss the relationship between the Laplace and the variational approximation and we show that for models with Gaussian priors and factorising likelihoods, the number of variational parameters is actually $\Ocal(N)$. The approach is applied to Gaussian process regression with non-Gaussian likelihoods.