Conformal prediction for uncertainties in nucleon-nucleon scattering

TL;DR

This work addresses uncertainty quantification in nucleon-nucleon scattering by applying conformal prediction as a distribution-free postprocessing tool to Bayesian posterior samples of observables such as total cross sections and phase shifts. It investigates three modeling scenarios—pointwise EFT coefficient sampling, Gaussian-process models for coefficients, and phase shifts from local N$^2$LO interactions—to construct distribution-free CP intervals, including conformalized quantile regression (CQR) corrections that yield adaptive, potentially asymmetric bands. The study demonstrates that CP provides reliable empirical coverage across energies and partial waves, remains robust to non-Gaussian features like skewness and heavy tails, and yields bands that agree with prior Bayesian credible intervals. This CP framework offers a practical, model-agnostic uncertainty-quantification tool with potential applications to neutron-star EOS, many-body calculations, and improved constraints on nucleon-nucleon interactions.</b>

Abstract

Conformal prediction is a distribution-free and model-agnostic uncertainty-quantification method that provides finite-sample prediction intervals with guaranteed coverage. In this work, for the first time, we apply conformal-prediction to generate uncertainty bands for physical observables in nuclear physics, such as the total cross section and nucleon-nucleon phase shifts. We demonstrate the method's flexibility by considering three scenarios: (i) a pointwise model, where expansion coefficients in chiral effective field theory are treated as random variables; (ii) a Gaussian-process model for the coefficients; and (iii) phase shifts at various energies and partial waves calculated using local interactions from chiral effective field theory. In each case, conformal-prediction intervals are constructed and validated empirically. Our results show that conformal prediction provides reliable and adaptive uncertainty bands even in the presence of non-Gaussian behavior, such as skewness and heavy tails. These findings highlight conformal prediction as a robust and practical framework for quantifying theoretical uncertainties.

Figures (18)

• Figure 1: Empirical quantile function constructed from samples drawn from a normal distribution with mean 2 and standard deviation 1. The curve maps quantile levels $\tau\in[0,1]$ to the corresponding quantiles $Q_Y(\tau)$. This function is the inverse of the cumulative distribution function.
• Figure 2: Quantile regression fits for different quantile levels $\tau\in \{0.05,0.25,0.5,0.75,0.95\}$ applied to synthetic data with heteroscedastic noise. Each fitted line estimates the $\tau$-th conditional quantile $Q_Y(\tau\mid X)$.The model $Y=2 + 3X+X\varepsilon$, where $\varepsilon\sim \mathcal{N}(2,2)$, generates data with increasing variance as $x$ increases.
• Figure 3: 95% naive CP interval for a toy dataset including 500 samples with heteroscedastic noise. The prediction model is a least squares regression fitted to the training data, and the shaded region represents the CP interval constructed using absolute residuals from the calibration data. The black line shows the fitted model. The empirical coverage is 0.9454.
• Figure 4: 95% CQR interval for a toy dataset including 500 samples with heteroscedastic noise. The prediction model is a quantile regression model trained to estimate the conditional quantiles of the data. The shaded region represents the conformalized quantile regression interval constructed by adjusting the initial quantile bounds using conformity scores from the calibration data. This approach produces asymmetric prediction bands that adapt to the variability of the data. The dashed line indicates the median prediction. The empirical coverage is 0.9499.
• Figure 5: Comparison of 95% naive CP intervals for three distributions: (a) Gaussian, (b) student-$t$ with four degrees of freedom, and (c) chi-squared with 4 degrees of freedom. Each subplot shows the histogram of 10000 sample values with the true distribution. The empirical coverage for each plot is calculated 0.9495, 0.9496, and 0.9499, respectively.