Moment-Based Variational Inference for Stochastic Differential Equations
This provides an improved inference method for researchers working with stochastic differential equations in fields like computational biology or finance, though it appears incremental as it builds on existing variational approaches.
The paper tackles the problem of variational inference for stochastic differential equations by constructing a variational process as a controlled version of the prior process and approximating the posterior using moment functions, reducing smoothing to a deterministic optimal control problem. The result is a method that allows for richer variational approximations extending to state-dependent diffusion terms, with the classical Gaussian process approximation recovered as a special case.
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.