  We consider fixed-interval smoothing problems for counter-adversarial
autonomous systems. An adversary deploys an autonomous filtering and control
system that i) measures our current state via a noisy sensor, ii) computes a
posterior estimate (belief) and iii) takes an action that we can observe. Based
on such observed actions and our knowledge of our state sequence, we aim to
estimate the adversary's past and current beliefs -- this forms a foundation
for predicting, and counteracting against, future actions. We derive the
optimal smoother for the adversary's beliefs (we treat the problem in a
Bayesian framework). Moreover, we demonstrate how the smoother can be computed
for discrete systems even though the corresponding backward variables do not
admit a finite-dimensional characterization. Finally, we illustrate our results
in numerical simulations.
