# Why Simple Quadrature is just as good as Monte Carlo

## TL;DR

This paper demonstrates that simple quadrature methods have expected variances comparable to or better than Monte Carlo, supported by theoretical proofs and simulations, challenging the notion that Monte Carlo is always superior for integration.

## Contribution

It establishes a statistical equivalence between deterministic quadrature and Monte Carlo sampling, leading to proofs that simple quadrature can outperform MC in variance and error bounds.

## Key findings

- Quadrature variance is less than or equal to Monte Carlo variance.
- Bayesian quadrature priors can be regularized for fair comparison with MC.
- Error standard deviations improve with sample size by a dimension-independent factor.

## Abstract

We motive and calculate Newton--Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton--Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor $\propto N^{-1/2}$. This dimension independent factor is validated in our simulations.