# The Trio Identity for Quasi-Monte Carlo Error

## TL;DR

This paper introduces the trio identity for Quasi-Monte Carlo error, linking integrand variation, sampling discrepancy, and confounding to better understand and reduce integration errors.

## Contribution

It formalizes the trio identity for Monte Carlo errors, highlighting the role of confounding and demonstrating how low discrepancy sampling improves accuracy.

## Key findings

- Confounding is a key factor in cubature error.
- Low discrepancy sampling reduces integration error.
- Rewriting the integrand can further decrease 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.