Rapid advances in designing cognitive and counter-adversarial systems have motivated the development of inverse Bayesian filters. In this setting, a cognitive `adversary' tracks its target of interest via a stochastic framework such as a Kalman filter (KF). The target or `defender' then employs another inverse stochastic filter to infer the forward filter estimates of the defender computed by the adversary. For linear systems, inverse Kalman filter (I-KF) has been recently shown to be effective in these counter-adversarial applications. In the paper, contrary to prior works, we focus on non-linear system dynamics and formulate the inverse unscented KF (I-UKF) to estimate the defender's state with reduced linearization errors. We then generalize this framework to an unknown system model by proposing reproducing kernel Hilbert space-based UKF (RKHS-UKF) to learn the system dynamics and estimate the state based on its observations. Our theoretical analyses to guarantee the stochastic stability of I-UKF and RKHS-UKF in the mean-squared sense shows that, provided the forward filters are stable, the inverse filters are also stable under mild system-level conditions. Our numerical experiments for several different applications demonstrate the state estimation performance of the proposed filters using recursive Cram\'{e}r-Rao lower bound as a benchmark.

翻译：认知系统与反对抗系统设计的快速发展推动了逆贝叶斯滤波器的研究。在该场景中，认知"对抗方"通过卡尔曼滤波器等随机框架追踪其感兴趣的目标。随后，目标或"防御方"采用另一种逆随机滤波器来推断对抗方计算出的防御方前向滤波估计。针对线性系统，逆卡尔曼滤波器近期已被证明可有效应用于此类反对抗场景。与先前研究不同，本文聚焦于非线性系统动力学，提出逆无迹卡尔曼滤波器以在降低线性化误差的同时估计防御方状态。进一步地，我们将该框架推广至未知系统模型，通过构建基于再生核希尔伯特空间的UKF来学习系统动力学并基于观测估计状态。理论分析保障了I-UKF和RKHS-UKF在均方意义上的随机稳定性，结果表明：当前向滤波器稳定时，在温和的系统级条件下逆滤波器同样保持稳定。针对多种不同应用的数值实验，以递归克拉美-罗下界为基准，验证了所提滤波器的状态估计性能。