PHYSICAL REVIEW E 94, 023305 (2016) Bayesian approach to the analysis of neutron Brillouin scattering data on liquid metals A. De Francesco, 1 , * E. Guarini, 2 U. Bafile, 3 F. Formisano, 1 and L. Scaccia 4 1 Consiglio Nazionale delle Ricerche, Istituto Officina dei Materiali, Operative Group in Grenoble (OGG), c/o Institut Laue Langevin, 71 Ave des Martyrs, BP 156 F-38042 Grenoble Cedex, France 2 Dipartimento di Fisica e Astronomia, Universit` a di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy 3 Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy 4 Dipartimento di Economia e Diritto, Universit ` a di Macerata, Via Crescimbeni 20, 62100 Macerata, Italy (Received 2 May 2016; published 8 August 2016) When the dynamics of liquids and disordered systems at mesoscopic level is investigated by means of inelastic scattering (e.g., neutron or x ray), spectra are often characterized by a poor definition of the excitation lines and spectroscopic features in general and one important issue is to establish how many of these lines need to be included in the modeling function and to estimate their parameters. Furthermore, when strongly damped excitations are present, commonly used and widespread fitting algorithms are particularly affected by the choice of initial values of the parameters. An inadequate choice may lead to an inefficient exploration of the parameter space, resulting in the algorithm getting stuck in a local minimum. In this paper, we present a Bayesian approach to the analysis of neutron Brillouin scattering data in which the number of excitation lines is treated as unknown and estimated along with the other model parameters. We propose a joint estimation procedure based on a reversible-jump Markov chain Monte Carlo algorithm, which efficiently explores the parameter space, producing a probabilistic measure to quantify the uncertainty on the number of excitation lines as well as reliable parameter estimates. The method proposed could turn out of great importance in extracting physical information from experimental data, especially when the detection of spectral features is complicated not only because of the properties of the sample, but also because of the limited instrumental resolution and count statistics. The approach is tested on generated data set and then applied to real experimental spectra of neutron Brillouin scattering from a liquid metal, previously analyzed in a more traditional way. DOI: 10.1103/PhysRevE.94.023305 I. INTRODUCTION The extraction of physical quantitative information from inelastic neutron scattering spectra is often a challenging task, especially when dealing with specific samples or anytime the scattering signal is very weak. These limiting conditions, together with the effect of both instrumental resolution and finite statistics [1,2], sometimes make the data analysis and the estimation of the relevant dynamical parameters similar to the amazing job of a sculptor who carves a beautiful and well defined figure from an unshaped block of marble. The physics we are interested in is often hidden behind and we cannot disclose it at a glance. When the collected spectra are poorly structured, in fact, two different issues arise. The first issue concerns the choice of the number of structures or peaks that are present in the measured spectra. The choice of the model to use could in some cases be based on strong beliefs, in some others just on a reasonable guess, and sometimes is just the best we can do to attempt any data analysis. This problem could also be dealt with by using various criteria, such as the Akaike information criterion (AIC) [3] or the Bayesian information criterion (BIC) [4]. These criteria, however, fail to provide a probability measure associated with the number of peaks and different criteria may lead to different conclusions. Once the number of these structures has been decided, peak parameters need to be estimated. This poses a second problem, namely, the fact that classical algorithms may fall and stop in * Corresponding author: defrance@ill.fr local extrema, producing results that strongly depend on the initial values of the parameters [5,6]. Different starting values may lead to different solutions and the discernment of the researcher may play a heavy role. These problems make it difficult to discover the physical phenomena which are underneath spectra encrusted with experimental noise and smeared by the unavoidably limited instrumental capabilities, and thus render the choice of an adequate model for the studied system more arbitrary. As an example, consider the case in which we are investigating the collective dynamics of a liquid in a whatsoever situation of confinement. Suppose that the dynamical behavior of the bulk liquid is well known by several and well assessed experimental works. Let us imagine that the spectroscopic data profile does not authorize any easy guess about where the excitation frequencies could be. It is quite natural, then, to provide the fitting algorithm with initial values corresponding to those known from the literature about the bulk liquid. However, the algorithm might converge to a local solution just close to the initial values we set, without adequately moving throughout the whole parameter space. If nothing is known about the sample in the explored confined conditions, then we could doubt whether the dynamical behavior of the confined liquid is effectively similar to that of the bulk fluid, or simply the algorithm failed to move away from the starting values. In such a situation, any speculation about dynamical behaviors under study could be fallacious or pushed beyond the evidence provided by the data. On the other hand, a random choice of the starting values could either determine convergence to a physically meaningless solution or prevent the algorithm from convergence, unless strict constraints on the parameters are 2470-0045/2016/94(2)/023305(14) 023305-1 ©2016 American Physical Society