Asymptotic optimality results for controlled sequential estimation

We consider the problem of sequential estimation of a random parameter under a controlled setting. Unlike traditional estimation problems, the collected observations depend on the used actions, which control the quality of the sensing process. At each time step, the decision maker chooses a control from a finite set of controls or decides to stop collecting measurements. The goal is to design an efficient causal control policy and a stopping rule and the efficiency is captured using the notion of asymptotic pointwise optimality (APO). This setup, in the context of sequential estimation for controlled parameter estimation was first considered in [1] for a special case where the distributions corresponding to different controls depend on uncommon parameters. In this paper, we extend the results in [1] to a more general case wherein the observation models under different controls could depend on common parameters. For this general setting, we propose a procedure consisting of a control policy and stopping rule, which is shown to be APO. In the process we identify and point out several applications, particularly in the area of active learning.