In this report, we revisit the work of Pilleboue et al. [2015], provid- ing a representation-theoretic derivation of the closed-form expres- sion for the expected value and variance in homogeneous Monte Carlo integration. We show that the results obtained for the vari- ance estimation of Monte Carlo integration on the torus, the sphere, and Euclidean space can be formulated as specific instances of a more general theory. We review the related representation theory and show how it can be used to derive a closed-form solution.
@techreport{kazhdan15rr,
author = {Kazhdan, Michael and Singh, Gurprit and Pilleboue, Adrien and Coeurjolly, David and Ostromoukhov, Victor},
hal_id = {hal-01259838},
institution = {LIRIS UMR CNRS 5205},
month = {March},
pdf = {https://hal.archives-ouvertes.fr/hal-01259838/file/1506.00021v1.pdf},
title = {Variance Analysis for Monte Carlo Integration: A Representation-Theoretic Perspective},
type = {Research Report},
url = {https://hal.archives-ouvertes.fr/hal-01259838},
year = {2015}
}