We address the problem of observation noise misspecification in Bayesian filtering of dynamical systems via recent advances in generalised Bayesian inference. Mis-match in tail decay between the true data generating process and an assumed observation model, often showing via frequent outliers, can strongly impact Bayesian updates and analysis in Kalman filtering. Existing approaches often employ detect-and-delete-schemes or covariance inflation to avoid assimilation of influential instances of mis-specification. In challenging settings where the analysis updates are barely sufficient to counteract the induced forecast uncertainty, these strategies may destabilize or struggle to provide reliable uncertainty quantification. We consider a novel Kalman filter adjusting information processing in the analysis step by employing diffusion score matching for inference to obtain robustness while maintaining well-quantified uncertainties. We provide theoretical properties of the diffusion score matching Kalman filter in linear Gaussian state space systems covering conjugacy and closed form parameter update in the analysis step, robustness, covariance stability, and tuning as well as high-dimensional consistency. We derive ensemble approximations via stochastic and deterministic coupling as well as implementing localization to obtain EnKF, ESRF and LETKF varieties. We evaluate the methods in appropriate simulation studies on target-tracking, the chaotic Lorenz 63 system and the Lorenz 96 system in 40 dimensions. Our insights highlight a critical trade-off between robustness and stability in Bayesian filtering. Methods employing generalized Bayesian inference can navigate this balance and improve data assimilation in challenging environments combining non-linear dynamics and potentially non-Gaussian observation noise.

翻译：我们通过广义贝叶斯推断的最新进展，解决了动态系统贝叶斯滤波中观测噪声误设定的问题。真实数据生成过程与假设的观测模型之间尾部衰减的不匹配（通常表现为频繁出现异常值）会严重影响卡尔曼滤波中的贝叶斯更新与分析。现有方法常采用“检测-剔除”方案或协方差膨胀来避免同化有影响力的误设定实例。在分析更新勉强足以抵消所诱发预报不确定性的挑战性场景中，这些策略可能不稳定或难以提供可靠的量化不确定性。我们提出一种新颖的卡尔曼滤波器，通过在分析步骤中采用扩散分数匹配进行推断来调整信息处理，从而在保持良好量化不确定性的同时获得鲁棒性。我们给出了扩散分数匹配卡尔曼滤波器在线性高斯状态空间系统中的理论性质，涵盖分析步骤中的共轭性与闭式参数更新、鲁棒性、协方差稳定性与调参，以及高维一致性。我们通过随机和确定性耦合推导集成近似，并实现局域化以得到EnKF、ESRF和LETKF变体。我们在目标跟踪、混沌洛伦兹63系统及40维洛伦兹96系统的适当仿真研究中评估了这些方法。我们的研究结果揭示了贝叶斯滤波中鲁棒性与稳定性之间的关键权衡。采用广义贝叶斯推断的方法可以平衡这一权衡，并在结合非线性动力学与潜在非高斯观测噪声的挑战性环境中改进数据同化。