Many approaches for conducting Bayesian inference on discretely observed diffusions involve imputing diffusion bridges between observations. This can be computationally challenging in settings in which the temporal horizon between subsequent observations is large, due to the poor scaling of algorithms for simulating bridges as observation distance increases. It is common in practical settings to use a blocking scheme, in which the path is split into a (user-specified) number of overlapping segments and a Gibbs sampler is employed to update segments in turn. Substituting the independent simulation of diffusion bridges for one obtained using blocking introduces an inherent trade-off: we are now imputing shorter bridges at the cost of introducing a dependency between subsequent iterations of the bridge sampler. This is further complicated by the fact that there are a number of possible ways to implement the blocking scheme, each of which introduces a different dependency structure between iterations. Although blocking schemes have had considerable empirical success in practice, there has been no analysis of this trade-off nor guidance to practitioners on the particular specifications that should be used to obtain a computationally efficient implementation. In this article we conduct this analysis and demonstrate that the expected computational cost of a blocked path-space rejection sampler applied to Brownian bridges scales asymptotically at a cubic rate with respect to the observation distance and that this rate is linear in the case of the Ornstein–Uhlenbeck process. Numerical experiments suggest applicability both of the results of our paper and of the guidance we provide beyond the class of linear diffusions considered.

对离散观察到的扩散进行贝叶斯推断的许多方法涉及在观察之间插入扩散桥。在后续观察之间的时间范围很大的设置中，这可能在计算上具有挑战性，因为随着观察距离的增加，用于模拟桥梁的算法的缩放比例很差。在实际设置中使用阻塞方案很常见，其中路径被分成（用户指定的）数量的重叠段，并使用 Gibbs 采样器依次更新段。将扩散桥的独立模拟替换为使用阻塞获得的独立模拟引入了固有的权衡：我们现在以在桥采样器的后续迭代之间引入依赖性为代价来估算较短的桥。由于有许多可能的方法来实现阻塞方案，这使情况变得更加复杂，每一种方法都在迭代之间引入了不同的依赖结构。尽管阻塞方案已经有相当多的经验在实践中取得成功，没有对这种权衡进行分析，也没有就应该用于获得计算效率实现的特定规范对从业者提供指导。在本文中，我们进行了这项分析并证明了应用于布朗桥的阻塞路径空间拒绝采样器的预期计算成本相对于观察距离以立方速率渐近缩放，并且该速率在 Ornstein 的情况下是线性的——乌伦贝克过程。数值实验表明我们论文的结果和我们提供的指导超出了所考虑的线性扩散类别的适用性。