# Estimating the Probability that a Function Observed with Noise is Convex

## TL;DR

This paper introduces a Bayesian sequential sampling method to estimate the probability that a noisy observed function is convex, utilizing Monte Carlo simulations and variance reduction techniques, with practical recommendations based on numerical experiments.

## Contribution

The paper presents a novel asymptotically consistent Bayesian approach for convexity testing of noisy functions, incorporating efficient variance reduction methods for posterior probability estimation.

## Key findings

- Conditional Monte Carlo outperforms other variance reduction methods.
- The proposed method is asymptotically consistent.
- Numerical experiments validate the effectiveness of the approach.

## Abstract

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.