Neglecting correlations leads to misestimated model errors in EFT predictions

TL;DR

This work demonstrates that in Bayesian EFT analyses, truncation (model) uncertainty and parametric uncertainty from LECs are not independent when predictions correlate with calibration data. By employing a toy EFT model and retaining the joint posterior of coefficients and the higher-order discrepancy term, the authors show that there is a meaningful anti-correlation between these uncertainties, which can substantially reduce the total predictive uncertainty compared to naive quadrature. The key contribution is formalizing and quantifying this anti-correlation, providing a practical framework for accurate uncertainty quantification in EFT predictions. The findings have important implications for EFT applications in nuclear physics, where under- or over-estimating uncertainties can affect interpretation and decision-making.

Abstract

Bayesian analyses of the convergence pattern of Effective Field Theories (EFTs) enable estimation of the uncertainty induced by a truncated expansion. When an EFT that has been calibrated to data is used to make a prediction this truncation uncertainty enters the posterior predictive distribution twice: directly from the finite-order calculation of the predicted quantity and indirectly through the posterior probability distributions of the EFT low-energy constants (LECs) determined by the calibration. In this work, we focus on the interplay of these two sources of uncertainty. We do this in the context of a toy EFT that we fit to pseudodata and use to make predictions. Direct EFT truncation uncertainty and LEC uncertainty are correlated in predictions when the predicted quantity is correlated with the observables used to fit the LECs. Here this results in the overall theoretical uncertainty in the EFT prediction being smaller than either the uncertainty induced by the truncation error or that stemming from the LECs alone.

Figures (5)

• Figure 1: The prediction for the value of the function $g(x)$ based on Bayesian linear regression carried out with a third-order polynomial, with a fourth-order term treated as model uncertainty and marginalized over. The prediction is conditioned on the data set $D_1$ from Refs. Schindler:2008fhWesolowski:2015fqa. The blue band represents the total model uncertainty of the prediction.
• Figure 2: Third-order polynomial prediction for the function $g(x)$, conditioned on the data set $D_1$---the same data set as in Fig. \ref{['fig:model_uncertainty']}. However, here the parameter uncertainty, $\sigma_{\text{param}}(x)$, which is depicted as the red band, and the truncation uncertainty, $\sigma_{\text{H.O.}}(x)$, depicted as the green band, are assessed separately. Comparing to Fig. \ref{['fig:model_uncertainty']}, we see that the actual uncertainty associated with the fourth-order term in the polynomial is smaller than either the parametric or truncation uncertainty bands shown here.
• Figure 3: The red line depicts the ratio of the parametric uncertainty $\sigma_{\rm param}(x)$ to the full uncertainty as a percentage. The green line depicts the ratio of the truncation uncertainty $a_{k+1} x^{k+1}$ to the full uncertainty.
• Figure 4: The full uncertainty of the model, computed by considering the anti-correlation as in Eq. (\ref{['eq:sigmacorrect']}), is shown in blue. The result obtained when this correlation is neglected, and so the truncation error and the parametric error are combined in quadrature, per Eq. \ref{['eq:sigmafull']}, is the markedly wider green band. The polynomial fit here is up to order $k=3$, and the quartic term is treated as a truncation error.
• Figure 5: The Pearson correlation coefficient $\rho(x)$ from Eq. \ref{['eq:pearson']} of the parametric and higher-order uncertainty as a function of $x$.