Title: The Trio Identity for Quasi-Monte Carlo Error,
Abstract: 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.