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, the 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 based on the unscented transform, or equivalently, statistical linearization technique. We then generalize this framework to unknown systems 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 show that, provided the forward filters are stable, the inverse filters are also stable under mild system-level conditions. We show that, despite being a suboptimal filter, our proposed I-UKF is a conservative estimator, i.e., I-UKF's estimated error covariance upper-bounds its true value. Our numerical experiments for several different applications demonstrate the estimation performance of the proposed filters using recursive Cram\'{e}r-Rao lower bound and non-credibility index (NCI).

翻译：认知与对抗系统设计的快速发展推动了逆贝叶斯滤波器的研究。在该框架中，认知型"对手"通过卡尔曼滤波器（KF）等随机方法跟踪其感兴趣的目标。目标或"防御者"随后采用另一种逆随机滤波器，推断对手计算得到的防御者前向滤波器估计值。对于线性系统，逆卡尔曼滤波器（I-KF）已被证明在此类对抗应用中具有有效性。与以往工作不同，本文聚焦于非线性系统动力学，基于无迹变换（即统计线性化技术）提出逆无迹卡尔曼滤波器（I-UKF）以估计防御者状态。进而将该框架推广至未知系统，提出基于再生核希尔伯特空间的UKF（RKHS-UKF），通过学习系统动力学并依据观测值进行状态估计。为保证I-UKF与RKHS-UKF在均方意义下的随机稳定性，理论分析表明：在前向滤波器稳定的条件下，逆滤波器在温和的系统级条件下同样保持稳定。尽管I-UKF属于次优滤波器，我们证明其具有保守估计特性，即I-UKF的估计误差协方差严格上界真实值。通过多种不同应用场景的数值实验，采用递归克拉美-罗下界与非可信度指数（NCI）验证了所提滤波器的估计性能。