Recent developments in counter-adversarial system research have led to the development of inverse stochastic filters that are employed by a defender to infer the information its adversary may have learned. Prior works addressed this inverse cognition problem by proposing inverse Kalman filter (I-KF) and inverse extended KF (I-EKF), respectively, for linear and non-linear Gaussian state-space models. However, in practice, many counter-adversarial settings involve highly non-linear system models, wherein EKF's linearization often fails. In this paper, we consider the efficient numerical integration techniques to address such nonlinearities and, to this end, develop inverse cubature KF (I-CKF) and inverse quadrature KF (I-QKF). We derive the stochastic stability conditions for the proposed filters in the exponential-mean-squared-boundedness sense. Numerical experiments demonstrate the estimation accuracy of our I-CKF and I-QKF with the recursive Cram\'{e}r-Rao lower bound as a benchmark.

翻译：对抗系统研究的最新进展推动了逆随机滤波器的发展，这类滤波器被防御方用于推断对手可能获取的信息。以往的研究通过提出逆卡尔曼滤波器（I-KF）和逆扩展卡尔曼滤波器（I-EKF），分别针对线性和非线性高斯状态空间模型解决了这一逆向认知问题。然而在实际应用中，许多对抗防御场景涉及高度非线性的系统模型，此时EKF的线性化方法常常失效。本文考虑采用高效的数值积分技术来处理此类非线性问题，并据此开发了逆Cubature卡尔曼滤波器（I-CKF）与逆Quadrature卡尔曼滤波器（I-QKF）。我们从指数均方有界的角度推导了所提滤波器的随机稳定性条件。数值实验以递归Cramér-Rao下界为基准，验证了I-CKF与I-QKF的估计精度。