  Bayesian filtering deals with computing the posterior distribution of the
state of a stochastic dynamic system given noisy observations. In this paper,
motivated by applications in counter-adversarial systems, we consider the
following inverse filtering problem: Given a sequence of posterior
distributions from a Bayesian filter, what can be inferred about the transition
kernel of the state, the observation likelihoods of the sensor and the measured
observations? For finite-state Markov chains observed in noise (hidden Markov
models), we show that a least-squares fit for estimating the parameters and
observations amounts to a combinatorial optimization problem with non-convex
objective. Instead, by exploiting the algebraic structure of the corresponding
Bayesian filter, we propose an algorithm based on convex optimization for
reconstructing the transition kernel, the observation likelihoods and the
observations. We discuss and derive conditions for identifiability. As an
application of our results, we illustrate the design of counter-adversarial
systems: By observing the actions of an autonomous enemy, we estimate the
accuracy of its sensors and the observations it has received. The proposed
algorithms are evaluated in numerical examples.
