Automatic differentiation for error analysis of Monte Carlo data

TL;DR

This paper introduces a novel method combining Automatic Differentiation with the $\Gamma$-method to accurately propagate errors in Monte Carlo data analysis, including complex observables and fit parameters.

Contribution

It presents a new approach that leverages Automatic Differentiation for precise error estimation in Monte Carlo data, extending to iterative and fit parameter calculations.

Findings

Exact error propagation for arbitrary observables

Improved error estimates in fit parameters

Implementation available in Fortran

Abstract

Automatic Differentiation (AD) allows to determine exactly the Taylor series of any function truncated at any order. Here we propose to use AD techniques for Monte Carlo data analysis. We discuss how to estimate errors of a general function of measured observables in different Monte Carlo simulations. Our proposal combines the $\Gamma$-method with Automatic differentiation, allowing exact error propagation in arbitrary observables, even those defined via iterative algorithms. The case of special interest where we estimate the error in fit parameters is discussed in detail. We also present a freely available fortran reference implementation of the ideas discussed in this work.