We study the problem of controlling a partially observed Markov decision process (POMDP) to either aid or hinder the estimation of its state trajectory by optimising the conditional entropy of the state trajectory given measurements and controls, a quantity we dub the smoother entropy. Our consideration of the smoother entropy contrasts with previous active state estimation and obfuscation approaches that instead resort to measures of marginal (or instantaneous) state uncertainty due to tractability concerns. By establishing novel expressions of the smoother entropy in terms of the usual POMDP belief state, we show that our active estimation and obfuscation problems can be reformulated as Markov decision processes (MDPs) that are fully observed in the belief state. Surprisingly, we identify belief-state MDP reformulations of both active estimation and obfuscation with concave cost and cost-to-go functions, which enables the use of standard POMDP techniques to construct tractable bounded-error (approximate) solutions. We show in simulations that optimisation of the smoother entropy leads to superior trajectory estimation and obfuscation compared to alternative approaches.
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