Half-way seminar in applied mathematics and statistics

Abstract: Stochastic differential equations (SDE) are employed in many areas of science as a powerful tool for modelling processes that are subject to random fluctuations. Bayesian inference for a large class of SDEs is challenging due to the analytic intractability of the likelihood function. Nevertheless, forward simulation via numerical methods is straightforward, motivating the use of approximate Bayesian computation (ABC). We propose a simulation scheme for SDE models that is based on processing the observation in both the forward and backward direction, effectively utilizing the information provided by the observed data. This leads to the simulation of sample paths that are consistent with the observations, thereby increasing the ABC acceptance rate. We additionally leverage partial exchangeability of Markov processes and employ invariant neural networks to learn the summary statistics that are needed in ABC. These are sequentially learned by exploiting a sequential Monte Carlo ABC sampler, which provides new training data at each iteration. Therefore, our novel contribution is a learning tool for SDE model parameters while simultaneously learning the summary statistics. Using synthetic data generated from the Chan-Karaolyi-Longstaff-Sanders SDE family, we show that our approach accelerates inference considerably, compared to standard (forward-only) methods, while preserving inference accuracy.