An efficient estimation of nested expectations without conditional sampling

TL;DR

This paper introduces a new Monte Carlo method using post-stratification for efficiently estimating nested expectations without conditional sampling, relying solely on joint samples of inner and outer variables.

Contribution

The proposed method enables estimation of nested expectations using only joint samples, improving efficiency and applicability over existing methods.

Findings

Outperforms nested Monte Carlo in efficiency

Comparable or better accuracy in numerical experiments

Requires only joint samples, not conditional samples

Abstract

Estimating nested expectations is an important task in computational mathematics and statistics. In this paper we propose a new Monte Carlo method using post-stratification to estimate nested expectations efficiently without taking samples of the inner random variable from the conditional distribution given the outer random variable. This property provides the advantage over many existing methods that it enables us to estimate nested expectations only with a dataset on the pair of the inner and outer variables drawn from the joint distribution. We show an upper bound on the mean squared error of the proposed method under some assumptions. Numerical experiments are conducted to compare our proposed method with several existing methods (nested Monte Carlo method, multilevel Monte Carlo method, and regression-based method), and we see that our proposed method is superior to the compared methods in terms of efficiency and applicability.