We develop a Bayesian inference method for 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 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 increasing the ABC acceptance rate. We additionally employ an invariant neural network, previously developed for Markov processes, to learn the summary statistics function required in ABC. The neural network is incrementally retrained by exploiting an ABC-SMC sampler, which provides new training data at each round. Since the SDEs simulation scheme differs from standard forward simulation, we propose a suitable importance sampling correction, which has the added advantage of guiding the parameters towards regions of high posterior density, especially in the first ABC-SMC round. Our approach achieves accurate inference and is about three 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 (Schl{\"o}gl) that is notoriously challenging for ABC methods, and a two dimensional biochemical reaction network.

翻译：我们针对离散观测的随机微分方程（SDEs）提出了一种贝叶斯推断方法。由于似然函数的解析不可处理性，大多数SDEs的推断具有挑战性。然而，通过数值方法进行前向模拟相对直接，这促使我们采用近似贝叶斯计算（ABC）。我们提出了一种基于序贯蒙特卡洛（SMC）前瞻策略和利用后向模拟的粒子平滑的SDEs条件模拟方案。该方案能够模拟与观测轨迹相一致的轨迹，从而提高ABC接受率。此外，我们采用先前为马尔可夫过程开发的不变神经网络来学习ABC所需的摘要统计量函数。该神经网络通过利用ABC-SMC采样器进行增量式再训练，该采样器在每一轮提供新的训练数据。由于SDEs模拟方案不同于标准的前向模拟，我们提出了一个合适的重采样校正方法，该方法具有引导参数向高后验密度区域移动的额外优势，尤其是在第一轮ABC-SMC中。我们的方法实现了精确推断，并且比标准（仅前向）ABC-SMC快约三倍。我们在五个模拟研究中展示了我们的方法，包括来自Chan-Karaolyi-Longstaff-Sanders SDE家族的三个例子、一个对ABC方法极具挑战性的随机双稳态模型（Schlögl），以及一个二维生化反应网络。