Bayesian Methods for Parameter Estimation in Effective Field Theories

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 chi-squared” 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 chi-squared 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.

💡 Research Summary

The paper presents a comprehensive Bayesian framework for extracting low‑energy constants (LECs) that encode short‑distance physics in effective field theories (EFTs). Traditional chi‑squared fitting, while straightforward, often becomes ambiguous when data are sparse, when the EFT expansion order is uncertain, or when some data lie outside the regime of convergence. To address these issues the authors combine Bayes’ theorem with the principle of maximum entropy to construct a “naturalness prior” that reflects the physical expectation that EFT parameters are of order unity (O(1)) in appropriate units. This prior is taken as a zero‑mean Gaussian with variance (R^{2}), where (R) quantifies the allowed deviation from exact unity. Crucially, (R) itself is treated as a hyper‑parameter and is marginalized over, thereby incorporating uncertainty about the precise scale of naturalness rather than fixing it arbitrarily.

A second source of model uncertainty is the truncation order (M) of the EFT expansion. Rather than committing to a single order, the authors assign prior probabilities to a set of plausible orders (e.g., (M=2,3,4)) and marginalize over (M). This procedure yields a posterior that automatically averages over the possible theoretical approximations, providing a systematic way to propagate truncation errors into the extracted parameters.

The full posterior for the parameters of interest (\mathbf{a}) is derived as
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