The Trio Identity for Quasi-Monte Carlo Error
Provides a unified theoretical framework for understanding and reducing Monte Carlo integration error, benefiting computational scientists and statisticians.
The paper introduces a trio identity for Monte Carlo error as the product of variation, discrepancy, and confounding, and shows how quasi-Monte Carlo methods reduce error via low discrepancy sampling. The confounding factor explains error decay rates differing from discrepancy.
Monte Carlo methods approximate integrals by sample averages of integrand values. The error of Monte Carlo methods may be expressed as a trio identity: the product of the variation of the integrand, the discrepancy of the sampling measure, and the confounding. The trio identity has different versions, depending on whether the integrand is deterministic or Bayesian and whether the sampling measure is deterministic or random. Although the variation and the discrepancy are common in the literature, the confounding is relatively unknown and under-appreciated. Theory and examples are used to show how the cubature error may be reduced by employing the low discrepancy sampling that defines quasi-Monte Carlo methods. The error may also be reduced by rewriting the integral in terms of a different integrand. Finally, the confounding explains why the cubature error might decay at a rate different from that of the discrepancy.