# Inelastic X-ray Scattering as a Probe of Terahertz Phonon Propagation in Nanoparticle Suspensions
## 2 The Pivotal Role of Bayesian Analysis  
Overall, IXS studies on nanostructured materials are still sporadic, primarily due to the complexity of their dynamic response and the usually weak and poorly resolved inelastic contributions to their spectral profile. We recall here that the shape of an IXS spectrum from a Soft Matter system is often relatively unstructured. A typical example is displayed in Figure 1 and refers to an aqueous suspension of gold nanoparticles (Au-NPs) measured at \(Q=3\) nm\({}^{-1}\). The plot compares the raw measurement with its best fitting model line shape and its elastic and inelastic contributions. As typical of these measurements, the spectral features of the raw line shape are not well pronounced and mutually resolved. However, the comparison with the model profiles helps us recognize that, at these relatively low \(Q\)s, the spectral shape is somehow reminiscent of the well-known Brillouin triplet typically observed in the hydrodynamic limit. The latter features a central peak--usually dominant in an IXS spectrum from a disordered system--and two broad shoulders symmetrically sitting on its wings. The elastic peak arises from all non-propagating processes occurring in the system over timescales too long to be adequately resolved by the measurement, such as relaxation or diffusion. These processes affect density fluctuations in the target system, dominating them at low frequencies, where their essentially elastic contribution becomes preponderant over the inelastic one. This trend becomes more pronounced upon increasing \(Q\), reflecting the decreasing ability of the system to support density wave propagation.  
Aside from these merely qualitative aspects, any detailed modeling of the spectral shape would require a firm theory of the IXS line shapes in the mesoscopic regime. Unfortunately, such a theory is currently lacking and usually replaced by suitable phenomenological generalizations of the exact hydrodynamic description valid at the lowest \(Q\)s and \(E\)s only. This generalized description demands, first and foremost, some assumption on the \(Q\)-dependent shape of the model profile and, in particular, on the number of spectral modes contributing to it. Aside from these inherent dilemmas, additional interpretation problems usually come from the \(\chi^{2}\) best fitting routines involving too many adjustable parameters, whose result often turns out to be model-dependent.  
To make the scenario even more puzzling, the insight typically sought from the line shape analyses of current IXS measurements is becoming increasingly informative and detailed. To unclutter this complicated matter, a reliable interpretation of the experimental spectral shape is highly desirable and would require, in the first place, evidence-based modeling recipes. In this endeavor, the use of Bayesian inference [19; 20] is highly beneficial:  
1. It can rate the plausibility of competing model options;
2. It privileges consistent models with fewer adjustable parameters;
3. It protects the numerical algorithm from remaining trapped in local, rather than global, \(\chi^{2}\) minima;
4. It may be implemented using minimally invasive external constraints;
5. It efficiently copes with the limited statistical accuracy of experimental results.  
Among various topics, this review will illustrate how a Bayesian inference model enables an informative and unbiased interpretation of line shape measurements. However, before going any further, it is helpful to describe some general principles of the Bayesian inference approach followed in the data analysis presented here.  
Figure 1: An example of an IXS spectrum from a dilute suspension of 15 nm sized gold nanoparticles, Au-NPs (circles), compared with the best fitting line shape and related elastic and inelastic contributions (see the legend).