Estimating the Probability that a Function Observed with Noise Is Convex
Nanjing Jian, Shane G. Henderson
INFORMS Journal on Computing · 2020 · Volume 32 · Issue 2
Abstract
Consider a real-valued function that can be only 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 is preferred.
Keywords
convexity detection, Bayesian sequential models, variance reduction, likelihood ratio, conditional Monte Carlo
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