An efficient estimation of nested expectations without conditional sampling

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 efficiently without taking samples of the inner random variable from conditional distribution given outer variable. This property provides advantage over many existing methods that it enables us only with dataset on pair variables drawn joint distribution. We show upper bound mean squared error proposed under some assumptions. Numerical experiments are conducted compare our several (nested method, multilevel regression-based method), see superior compared terms efficiency applicability.