Low-Variance Importance Sampling for Discretely Observed Stochastic Differential Equations

Abstract: Stochastic differential equations (SDEs) are commonly used to model dynamical systems of interest. When such systems are observed at discrete times, they give rise to continuous–discrete state space models, where the main tasks are Bayesian inference of the latent state (filtering and smoothing) as well as computation of the observation likelihood (for model comparison). In linear and Gaussian settings these problems can be solved efficiently with recursive algorithms like the Kalman filter, but in nonlinear cases Kalman-type solutions require approximations and lead to biased solutions.

To overcome these problems, alternative solutions include sequential importance sampling (SIS), or perhaps most commonly, sequential importance resampling (SIR), also known as particle filters. These methods are unbiased and converge weakly to the correct solution as the number of particles goes to infinity. However, the efficiency of these methods depends greatly on the choice of importance distribution, as poor choices lead to high-variance importance weights and particle degeneracy. The design of good importance sampling distributions is therefore of crucial importance. For smoothing, filtering, and estimation of the observation likelihood, the optimal importance distribution is given by the smoothing distribution, which is the focus of this work.

By Doob’s h-transform, the smoothing distribution can be characterized as the law of a controlled SDE that differs from the unconditional one only by an additional drift term that steers trajectories toward future observations. In this work, we approximate this control term using neural networks, yielding a tractable approximation of the smoothing distribution that can be corrected with low-variance importance weights. The model is trained using divergence-based objectives, including the Kullback–Leibler divergence, and evaluated in terms of effective sample size and variance of likelihood estimates. This approach reduces variance in importance sampling and improves the efficiency of inference in nonlinear SDE models.

machine learningprobabilitystatistics theory

Audience: researchers in the discipline

Gothenburg statistics seminar

Series comments: Gothenburg statistics seminar is open to the interested public, everybody is welcome. It usually takes place in MVL14 (http://maps.chalmers.se/#05137ad7-4d34-45e2-9d14-7f970517e2b60, see specific talk). Speakers are asked to prepare material for 35 minutes excluding questions from the audience.