Bayesian methods for parameter estimation in effective field theories

Abstract

We demonstrate and explicate Bayesian methods for fitting the parameters that encode the impact of short-distance physics on observables in effective field theories (EFTs). We use Bayes' theorem together with the principle of maximum entropy to account for the prior information that these parameters should be natural, i.e. O(1) in appropriate units. Marginalization can then be employed to integrate the resulting probability density function (pdf) over the EFT parameters that are not of specific interest in the fit. We also explore marginalization over the order of the EFT calculation, M , and over the variable, R, that encodes the inherent ambiguity in the notion that these parameters are O(1). This results in a very general formula for the pdf of the EFT parameters of interest given a data set, D. We use this formula and the simpler "augmented χ 2 " in a toy problem for which we generate pseudo-data. These Bayesian methods, when used in combination with the "naturalness prior", facilitate reliable extractions of EFT parameters in cases where χ 2 methods are ambiguous at best. We also examine the problem of extracting the nucleon mass in the chiral limit, M 0 , and the nucleon sigma term, from pseudo-data on the nucleon mass as a function of the pion mass. We find that Bayesian techniques can provide reliable information on M 0 , even if some of the data points used for the extraction lie outside the region of applicability of the EFT.

Exploring Bayesian parameter estimation for chiral effective field theory using nucleon–nucleon phase shifts

“…Given the LECs, the prediction is exactly known, while we have shown that the theoretical uncertainty is a multivariate Gaussian. Assuming that the experimental noise is also a Gaussian For the special case of the fully correlated coefficients and a finite k max , we could instead integrate in each (univariate) unknown coefficient [30,19] The form of (B.42) is potentially advantageous when k max − k is small and Σ exp is diagonal, since the size of the matrix inversion for A is only (k max − k) × (k max − k) rather than N d × N d as in (B.33).…”

Flavor decomposition of the pion-nucleon σ-term

“…We therefore use a so-called prior-fit and modify our χ 2 to, see e.g. [53], Overall the prior-fit method behaves stable, since the minimum is not as flat as without the additional terms and all 19 LECs can be fitted quite precisely. However, we have to pay attention to choosing the errors ∆b i and ∆d i .…”

One-loop analysis of the interactions between charmed mesons and Goldstone bosons

“…Therefore, we try to constrain all the LECs to natural values following ref. [57] which discusses the use of the Bayesian method in effective field theories. Following Table 5.…”