The Monte Carlo replica method: investigating the effects of non-linearity

TL;DR

This paper rigorously analyzes the Monte Carlo replica method in high-energy physics, revealing its limitations in non-linear models and comparing its uncertainty estimates with Bayesian posteriors.

Contribution

It provides the first rigorous derivation of parameter distributions from the Monte Carlo replica method and assesses its accuracy in non-linear contexts.

Findings

Distributions match Bayesian posteriors in linear models

Discrepancies arise in non-linear models

Uncertainty estimates differ significantly in non-linear cases

Abstract

This paper presents an in-depth mathematical analysis of the Monte Carlo replica method, commonly used in global fitting studies within the high-energy physics theory field. For the first time, we offer a rigorous derivation of the parameter distributions resulting from this method, demonstrating that, while they align with Bayesian posteriors in linear models, they deviate in non-linear cases. We then numerically assess this discrepancy in a phenomenologically important context: fitting SMEFT Wilson coefficients. Our findings reveal that when non-linearity plays a significant role, the uncertainty estimates for key quantities differ between the two approaches.