Statistical Inference for Approximate Bayesian Optimal Design

This paper studies a generic Bayesian optimal design formulation with chance constraints, where the decision variable lies in a separable, reflexive Banach space. This setting covers a gamut of simulation and modeling problems that we illustrate through two example problem formulations. The posterior objective cannot be computed, in general, and it is necessary to use approximate Bayesian inference. Sampling-based approximate inference, however, introduces significant variance and, in general, leads to non-convex approximate feasible sets, even when the original problem is convex. In this paper, we use variational Bayesian approximations that introduce no variance and retain the convexity of the feasibility set, subject to easily satisfied regularity conditions on the approximate posterior, albeit at the expense of a much larger bias. Our main results, therefore, establish large sample asymptotic consistency of the optimal solutions and optimal value of this approximate Bayesian optimal design formulation.