  We study the problem of controlling a partially observed Markov decision
process (POMDP) to either aid or hinder the estimation of its state trajectory.
We encode the estimation objectives via the smoother entropy, which is the
conditional entropy of the state trajectory given measurements and controls.
Consideration of the smoother entropy contrasts with previous approaches that
instead resort to marginal (or instantaneous) state entropies due to
tractability concerns. By establishing novel expressions for the smoother
entropy in terms of the POMDP belief state, we show that both the problems of
minimising and maximising the smoother entropy in POMDPs can surprisingly be
reformulated as belief-state Markov decision processes with concave cost and
value functions. The significance of these reformulations is that they render
the smoother entropy a tractable optimisation objective, with structural
properties amenable to the use of standard POMDP solution techniques for both
active estimation and obfuscation. Simulations illustrate that optimisation of
the smoother entropy leads to superior trajectory estimation and obfuscation
compared to alternative approaches.
