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Perturbation Theory for Variational Inference

The variational approximation is known to underestimate the true variance of a probability measure, like a posterior distribution. In latent variable models whose parameters are permutation-invariant with respect to the likelihood, the variational approximation might self-prune and ignore its parameters. We view the variational free energy and its accompanying evidence lower bound as a first-order term from a perturbation of the true log partition function and derive a power series of corrections. We sketch by means of a “variational matrix factorization” example how a further term could correct for predictions from a self-pruned approximation.