  Variational inference has become one of the most widely used methods in
latent variable modeling. In its basic form, variational inference employs a
fully factorized variational distribution and minimizes its KL divergence to
the posterior. As the minimization can only be carried out approximately, this
approximation induces a bias. In this paper, we revisit perturbation theory as
a powerful way of improving the variational approximation. Perturbation theory
relies on a form of Taylor expansion of the log marginal likelihood, vaguely in
terms of the log ratio of the true posterior and its variational approximation.
While first order terms give the classical variational bound, higher-order
terms yield corrections that tighten it. However, traditional perturbation
theory does not provide a lower bound, making it inapt for stochastic
optimization. In this paper, we present a similar yet alternative way of
deriving corrections to the ELBO that resemble perturbation theory, but that
result in a valid bound. We show in experiments on Gaussian Processes and
Variational Autoencoders that the new bounds are more mass covering, and that
the resulting posterior covariances are closer to the true posterior and lead
to higher likelihoods on held-out data.
