  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.
