Particle Smoothing in Continuous Time: A Fast Approach via Density Estimation

We consider the particle smoothing problem for state-space models where the transition density is not available in closed form, in particular for continuous-time, nonlinear models expressed via stochastic differential equations (SDEs). Conventional forward-backward and two-filter smoothers for the particle filter require a closed-form transition density, with the linear-Gaussian Euler-Maruyama discretization usually applied to the SDEs to achieve this. We develop a pair of variants using kernel density approximations to relieve the dependence, and in doing so enable use of faster and more accurate discretization schemes such as Runge-Kutta. In addition, the new methods admit arbitrary proposal distributions, providing an avenue to deal with degeneracy issues. Experimental results on a functional magnetic resonance imaging (fMRI) deconvolution task demonstrate comparable accuracy and significantly improved runtime over conventional techniques.