  Consider a real-valued function that can only be observed with stochastic
noise at a finite set of design points within a Euclidean space. We wish to
determine whether there exists a convex function that goes through the true
function values at the design points. We develop an asymptotically consistent
Bayesian sequential sampling procedure that estimates the posterior probability
of this being true. In each iteration, the posterior probability is estimated
using Monte Carlo simulation. We offer three variance reduction methods --
change of measure, acceptance-rejection, and conditional Monte Carlo. Numerical
experiments suggest that the conditional Monte Carlo method should be
preferred.
