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：最近对抗系统研究的发展引发了逆随机滤波器的开发，这些滤波器由防御者使用，以推断对手可能学到的信息。以前的工作通过提出逆卡尔曼滤波器（I-KF）和逆扩展KF（I-EKF）分别用于线性和非线性高斯状态空间模型来解决这种反认知问题。但是，在实践中，许多对抗性设置涉及高度非线性的系统模型，在这种情况下，EKF的线性化经常失败。在本文中，我们考虑使用有效的数值积分技术来解决这样的非线性问题，并为此开发逆立方测度KF（I-CKF）和逆积分KF（I-QKF）。我们以指数-均方有界性的意义为基础，推导了所提出的滤波器的随机稳定性条件。数值实验展示了我们的I-CKF和I-QKF的估计精度，并以递归Cram\'{e}r-Rao下界为基准。