Towards Data-Conditional Simulation for ABC Inference in Stochastic Differential Equations

Abstract

We develop a Bayesian inference method for the parameters of discretely-observed stochastic differential equations (SDEs). Inference is challenging for most SDEs, due to the analytical intractability of the likelihood function. Nevertheless, forward simulation via numerical methods is straightforward, motivating the use of approximate Bayesian computation (ABC). We propose a computationally efficient “data-conditional” simulation scheme for SDEs that is based on lookahead strategies for sequential Monte Carlo (SMC) and particle smoothing using backward simulation. This leads to the simulation of trajectories that are consistent with the observed trajectory, thereby considerably increasing the acceptance rate in an ABC-SMC sampler. As a result, our procedure rapidly guides the parameters towards regions of high posterior density, especially in the first ABC-SMC round. We additionally construct a sequential scheme to learn the ABC summary statistics, by employing an invariant neural network, previously developed for Markov processes, that is incrementally retrained during the run of the ABC-SMC sampler. Our approach achieves accurate inference and is about three times faster (and in some cases even 10 times faster) than standard (forward-only) ABC-SMC. We illustrate our method in five simulation studies, including three examples from the Chan–Karaolyi–Longstaff–Sanders SDE family, a stochastic bi-stable model (Schlögl) that is notoriously challenging for ABC methods, and a two dimensional biochemical reaction network.