# Markov Chain Monte Carlo technics applied to Parton Distribution Functions determination: proof of concept

## TL;DR

This paper introduces a novel approach using Markov Chain Monte Carlo methods, specifically Hybrid Monte Carlo, to determine Parton Distribution Functions, providing an alternative to traditional chi-squared minimization and offering new insights into PDF uncertainties.

## Contribution

The paper demonstrates the feasibility of applying MCMC, especially Hybrid Monte Carlo, to PDFs determination, addressing high-dimensional challenges and improving uncertainty estimation.

## Key findings

- MCMC can successfully extract PDFs and their uncertainties.
- Hybrid Monte Carlo circumvents high-dimensional acceptance issues.
- Feasibility study confirms potential of MCMC in global PDF analysis.

## Abstract

We present a new procedure to determine Parton Distribution Functions (PDFs), based on Markov Chain Monte Carlo (MCMC) methods. The aim of this paper is to show that we can replace the standard $\chi^2$ minimization by procedures grounded on Statistical Methods, and on Bayesian inference in particular, thus offering additional insight into the rich field of PDFs determination. After a basic introduction to these technics, we introduce the algorithm we have chosen to implement -- namely Hybrid (or Hamiltonian) Monte Carlo. This algorithm, initially developed for Lattice QCD, turns out to be very interesting when applied to PDFs determination by global analyses; we show that it allows to circumvent the difficulties due to the high dimensionality of the problem, in particular concerning the acceptance. A first feasibility study is performed and presented, which indicates that Markov Chain Monte Carlo can successfully be applied to the extraction of PDFs and of their uncertainties.