This paper considers data-driven chance-constrained stochastic optimization
problems in a Bayesian framework. Bayesian posteriors afford a principled
mechanism to incorporate data and prior knowledge into stochastic optimization
problems. However, the computation of Bayesian posteriors is typically an
intractable problem, and has spawned a large literature on approximate Bayesian
computation. Here, in the context of chance-constrained optimization, we focus
on the question of statistical consistency (in an appropriate sense) of the
optimal value, computed using an approximate posterior distribution. To this
end, we rigorously prove a frequentist consistency result demonstrating the
convergence of the optimal value to the optimal value of a fixed, parameterized
constrained optimization problem. We augment this by also establishing a
probabilistic rate of convergence of the optimal value. We also prove the
convex feasibility of the approximate Bayesian stochastic optimization problem.
Finally, we demonstrate the utility of our approach on an optimal staffing
problem for an M/M/c queueing model