Bootstrap-determined p-values in Lattice QCD

TL;DR

This paper introduces a bootstrap-based method to accurately compute p-values for lattice QCD data, especially when traditional methods are inapplicable, enhancing hypothesis testing in complex statistical scenarios.

Contribution

It develops a general bootstrap approach to determine p-values for lattice QCD data, applicable even with uncorrelated fits and minimal assumptions, improving statistical inference methods.

Findings

Bootstrap method accurately determines p-values in lattice QCD.

Applicable to uncorrelated fits where traditional methods fail.

Quantifies impact of finite sample size on p-value estimation.

Abstract

We present a general method to determine the probability that stochastic Monte Carlo data, in particular those generated in a lattice QCD calculation, would have been obtained were that data drawn from the distribution predicted by a given theoretical hypothesis. Such a probability, or p-value, is often used as an important heuristic measure of the validity of that hypothesis. The proposed method offers the benefit that it remains usable in cases where the standard Hotelling $T^2$ methods based on the conventional $\chi^2$ statistic do not apply, such as for uncorrelated fits. Specifically, we analyze a general alternative to the correlated $\chi^2$ statistic referred to as $q^2$, and show how to use the bootstrap as a data-driven method to determine the expected distribution of $q^2$ for a given hypothesis with minimal assumptions. This distribution can then be used to determine the p-value for a fit to the data. We also describe a bootstrap approach for quantifying the impact upon this p-value of estimating population parameters from a single ensemble of $N$ samples. The overall method is accurate up to a $1/N$ bias which we do not attempt to quantify.