Asymptotic Optimality of Sequential Sampling Policies for Bayesian Information Collection

We consider adaptive sequential sampling policies in a Bayesian framework. Under the assumptions that the sampling distribution is from an exponential family and that the number of distinct measurement types is finite, we give sufficient conditions for an adaptive sampling policy to achieve asymptotic optimality. Here, asymptotic optimality is understood to mean that the limit of the expected loss under the given sampling policy as the number of measurements allowed grows to infinity attains the minimum over all possible sampling policies. This property is important because it ensures convergence in the limit for sophisticated policies designed to maximize performance over the short-term. We then apply these sufficient conditions to show asymptotic optimality of three previously proposed ranking and selection policies: OCBA for linear loss, LL(S), and LL(1). We also show how this sufficient condition may be generally applied to a broad class of knowledge-gradient policies.