Moment-Based Variational Inference for Stochastic Differential Equations

TL;DR

This paper introduces a novel moment-based variational inference method for stochastic differential equations, transforming the smoothing problem into a deterministic optimal control task with enhanced approximation capabilities.

Contribution

It proposes a controlled process approach using moment functions and Fisher information, enabling richer variational approximations including state-dependent diffusions.

Findings

Enables variational inference with complex diffusion terms

Reduces smoothing to a deterministic optimal control problem

Recovers classical Gaussian process approximation as a special case

Abstract

Existing deterministic variational inference approaches for diffusion processes use simple proposals and target the marginal density of the posterior. We construct the variational process as a controlled version of the prior process and approximate the posterior by a set of moment functions. In combination with moment closure, the smoothing problem is reduced to a deterministic optimal control problem. Exploiting the path-wise Fisher information, we propose an optimization procedure that corresponds to a natural gradient descent in the variational parameters. Our approach allows for richer variational approximations that extend to state-dependent diffusion terms. The classical Gaussian process approximation is recovered as a special case.