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In this paper, we consider the problem of controlling a partially observedMarkov decision process (POMDP) in order to actively estimate its statetrajectory over a fixed horizon with minimal uncertainty. We pose a novelactive smoothing problem in which the objective is to directly minimise thesmoother entropy, that is, the conditional entropy of the (joint) statetrajectory distribution of concern in fixed-interval Bayesian smoothing. Ourformulation contrasts with prior active approaches that minimise the sum ofconditional entropies of the (marginal) state estimates provided by Bayesianfilters. By establishing a novel form of the smoother entropy in terms of thePOMDP belief (or information) state, we show that our active smoothing problemcan be reformulated as a (fully observed) Markov decision process with a valuefunction that is concave in the belief state. The concavity of the valuefunction is of particular importance since it enables the approximate solutionof our active smoothing problem using piecewise-linear function approximationsin conjunction with standard POMDP solvers. We illustrate the approximatesolution of our active smoothing problem in simulation and compare itsperformance to alternative approaches based on minimising marginal stateestimate uncertainties.

Active Trajectory Estimation for Partially Observed Markov Decision Processes via Conditional Entropy