Sequential detection of convexity from noisy function evaluations

Consider a real-valued function that can only be evaluated with error. Given estimates of the function values from simulation on a finite set of points, we seek a procedure to detect convexity or non-convexity of the true function restricted to those points. We review an existing frequentist hypothesis test, and introduce a sequential Bayesian procedure. Our Bayesian procedure applies for both independent sampling and sampling with common random numbers, with known or unknown sampling variance. In each iteration, we collect a set of samples and update a posterior distribution on the function values, and use that as the prior belief in our next iteration. We then approximate the probability that the function is convex based on the posterior using Monte Carlo simulation.