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

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.