  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).
