Bayesian filtering is a key tool in many problems that involve the online processing of data, including data assimilation, optimal control, nonlinear tracking and others. Unfortunately, the implementation of filters for nonlinear, possibly high-dimensional, dynamical systems is far from straightforward, as computational methods have to meet a delicate trade-off involving stability, accuracy and computational cost. In this paper we investigate the design, and theoretical features, of constrained Bayesian filters for state space models. The constraint on the filter is given by a sequence of compact subsets of the state space that determines the sources and targets of the Markov transition kernels in the dynamical model. Subject to such constraints, we provide sufficient conditions for filter stability and approximation error rates with respect to the original (unconstrained) Bayesian filter. Then, we look specifically into the implementation of constrained filters in a continuous-discrete setting where the state of the system is a continuous-time stochastic Itô process but data are collected sequentially over a time grid. We propose an implementation of the constraint that relies on a data-driven modification of the drift of the Itô process using barrier functions, and discuss the relation of this scheme with methods based on the Doob $h$-transform. Finally, we illustrate the theoretical results and the performance of the proposed methods in computer experiments for a partially-observed stochastic Lorenz 96 model.

翻译：贝叶斯滤波是许多涉及数据在线处理问题的关键工具，包括数据同化、最优控制、非线性跟踪等。然而，针对非线性且可能高维的动态系统实现滤波器远非易事，因为计算方法必须在稳定性、精度和计算成本之间达成微妙的平衡。本文研究了状态空间模型中约束贝叶斯滤波器的设计及其理论特性。滤波器的约束由状态空间的一系列紧致子集给出，这些子集决定了动态模型中马尔可夫转移核的源与目标。在此类约束条件下，我们为滤波器稳定性以及相对于原始（无约束）贝叶斯滤波器的近似误差率提供了充分条件。随后，我们具体探讨了约束滤波器在连续-离散场景下的实现，其中系统状态为连续时间随机Itô过程，但数据在时间网格上顺序采集。我们提出了一种约束实现方案，该方案基于使用障碍函数对Itô过程漂移项进行数据驱动的修正，并讨论了此方案与基于Doob $h$-变换的方法之间的关系。最后，我们通过针对部分观测的随机Lorenz 96模型的计算机实验，阐述了理论结果并展示了所提方法的性能。