Sampling, variational Bayesian inference, and conditioned stochastic differential equations

We consider the problem of sampling and statistical inference in probabilistic generative models, where the latent object is a finite-dimensional diffusion process. In general, it is difficult to obtain exact expressions for the log-likelihood, so one has to resort to so-called variational inference, where a change of measure is used to come up with a tractable upper bound. We first show, using W. Fleming’s logarithmic transformation, that the problem of constructing a variational approximation to the log-likelihood can be interpreted as an optimal control problem, where the choice of a variational approximation amounts to adding a drift to the original diffusion. We then analyze this class of control problems using the formalism of conditioned stochastic differential equations due to F. Baudoin. We discuss the relation of this problem to entropic optimal transport and to the stochastic maximum principle.