Computing the variance of a conditional expectation via non-nested Monte Carlo

TL;DR

This paper introduces unbiased non-nested Monte Carlo estimators using the pick-freeze scheme to efficiently compute the variance and higher moments of a conditional expectation, improving upon nested methods.

Contribution

It presents a novel non-nested Monte Carlo approach for variance estimation of conditional expectations, extending to higher moments, and offers an alternative to existing nested estimators.

Findings

Unbiased non-nested estimators are constructed using the pick-freeze scheme.

The approach extends to higher order moments of conditional expectations.

The method provides computational advantages over nested Monte Carlo estimators.

Abstract

Computing the variance of a conditional expectation has often been of importance in uncertainty quantification. Sun et al. has introduced an unbiased nested Monte Carlo estimator, which they call $1\frac{1}{2}$-level simulation since the optimal inner-level sample size is bounded as the computational budget increases. In this letter we construct unbiased non-nested Monte Carlo estimators based on the so-called pick-freeze scheme due to Sobol'. An extension of our approach to compute higher order moments of a conditional expectation is also discussed.