BUQEYE Model for EFT Uncertainty Quantification

• BUQEYE model is a Bayesian framework that uses Gaussian processes to model dimensionless coefficients and quantify EFT truncation errors.
• It extracts empirical order-by-order convergence data to infer hyperparameters and generate statistically calibrated uncertainty bands.
• The model also integrates projection-based emulators for fast surrogate computations and adapts to nonstationary behaviors in nucleon-nucleon interactions.

Searching arXiv for BUQEYE-related papers to ground the article in the latest research. BUQEYE, expanded by the collaboration as Bayesian Uncertainty Quantification: Errors in Your EFT, denotes a Bayesian framework for uncertainty quantification in effective field theory (EFT), especially for correlated truncation errors in low-energy nuclear physics, and more broadly a research program that also includes projection-based emulators for fast surrogate modeling (Drischler et al., 2022). In its truncation-error formulation, the central hypothesis is that dimensionless coefficient functions extracted from order-by-order EFT calculations can be treated as independent draws from a common Gaussian process (GP), enabling lower-order results to inform predictive distributions for omitted higher-order terms (Millican et al., 2024). Subsequent studies have used this model to assess convergence patterns across modern nucleon-nucleon potentials, infer breakdown scales such as $\Lambda_b$, and construct statistically calibrated uncertainty bands for observables including Wolfenstein amplitudes (Millican et al., 25 Aug 2025, McClung et al., 16 Jan 2025).

1. Definition and scope

In the nuclear-physics literature represented here, the BUQEYE model is a correlated EFT truncation-error model in which order-by-order observable calculations are recast in terms of dimensionless coefficients and those coefficients are modeled probabilistically by a GP (Millican et al., 2024). The collaboration’s guide to projection-based emulators places this statistical work within a larger BUQEYE framework concerned with Bayesian uncertainty quantification and computational acceleration for many-body and scattering calculations (Drischler et al., 2022).

The core application is the EFT expansion of an observable. One writes the prediction at truncation order $k$ as

$y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$

with $y_{\rm ref}(x)$ a reference scale, $c_n(x)$ dimensionless coefficients, and $Q(x)$ the expansion parameter (Millican et al., 2024). The truncation error is then

$$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$

so the statistical problem reduces to learning the size and correlation structure of the unknown higher-order coefficients from the known lower-order ones (Millican et al., 2024).

A related but distinct BUQEYE usage appears in the literature on Wolfenstein amplitudes, where the same GP machinery is applied to angle-correlated truncation uncertainties for the five independent on-shell nucleon-nucleon amplitude functions (McClung et al., 16 Jan 2025). This suggests that “BUQEYE model” is best understood not as a single narrow algorithm, but as a family of closely related Bayesian GP constructions for EFT-order uncertainty propagation.

2. Bayesian hypothesis and mathematical structure

The defining BUQEYE hypothesis is that once the observable has been normalized by a reference scale and the powers of the EFT expansion parameter have been factored out, the coefficients behave like independent draws from the same GP:

$c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$

In the formulation used for nucleon-nucleon observables, the correlation kernel is the squared-exponential kernel,

$$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$

where $\sigma^2$ is the marginal variance and $k$0 is a diagonal matrix of correlation-length scales such as $k$1 in energy and $k$2 in angle (Millican et al., 2024).

The same structure appears in the Wolfenstein-amplitude analysis, specialized to a one-dimensional input variable $k$3:

$k$4

with

$k$5

There, the hyperparameters are the signal variance $k$6, the correlation length $k$7, and the EFT breakdown scale $k$8 entering $k$9 (McClung et al., 16 Jan 2025).

The observable expansion is paired with a specific expansion parameter. One common choice is

$y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$0

with $y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$1 a characteristic momentum, $y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$2 a soft scale, and $y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$3 the EFT breakdown scale (Millican et al., 2024). In the Wolfenstein-amplitude study, the expansion parameter is instead written

$y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$4

with the same role assigned to $y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$5 (McClung et al., 16 Jan 2025).

Because the omitted terms are sums of GP-distributed coefficient functions weighted by powers of $y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$6, BUQEYE induces a GP—or, after marginalizing over variance parameters, a Student-$y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$7 process—for the truncation error itself (Millican et al., 25 Aug 2025). In the 2024 assessment paper, the predictive covariance for the missing terms at order $y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$8 is

$y_k(x)=y_{\rm ref}(x)\sum_{n=0}^k c_n(x)\,Q^n(x),$9

which directly encodes correlated uncertainty bands across kinematics (Millican et al., 2024).

3. Inference, diagnostics, and workflow

BUQEYE proceeds by extracting empirical coefficients from order-by-order calculations. Given predictions $y_{\rm ref}(x)$0, one computes

$y_{\rm ref}(x)$1

with $y_{\rm ref}(x)$2 (Millican et al., 2024). These extracted coefficient curves are then used to fit GP hyperparameters by conditioning the prior on a training set and scoring candidate length scales via the GP marginal likelihood, or its Student-$y_{\rm ref}(x)$3 analog after integrating out $y_{\rm ref}(x)$4 (Millican et al., 2024).

The published workflow is explicit. It begins with data preparation on a dense grid, choice of $y_{\rm ref}(x)$5 and $y_{\rm ref}(x)$6, coefficient extraction, and a train–test split. Hyperparameter inference is followed by a stationarity check using validation diagnostics. If the diagnostics pass, one performs scale inference over $y_{\rm ref}(x)$7, marginalizes to obtain posteriors for the physical scales, and then refits the GP to compute final truncation-error bands (Millican et al., 2024).

Three diagnostics recur across the BUQEYE literature. The first is a Mahalanobis-distance diagnostic, in which

$y_{\rm ref}(x)$8

should follow $y_{\rm ref}(x)$9 under the GP model, with outliers indicating mis-sized variance or length scale (Millican et al., 2024). The second is the pivoted-Cholesky diagnostic, in which

$c_n(x)$0

should have standard normal components if the covariance model is correct; funneling or trumpeting patterns reveal under- or overestimated length scales (Millican et al., 2024). The third is the credible-interval coverage or “weather” plot, which compares empirical coverage to nominal probability and is also used in later convergence assessments (Millican et al., 2024, Millican et al., 25 Aug 2025).

A concise summary of the standard BUQEYE analysis pipeline is given below.

Step	Operation	Purpose
1	Compute $c_n(x)$1 on a grid	Assemble order-by-order EFT data
2	Choose $c_n(x)$2 and $c_n(x)$3	Define dimensionless coefficients
3	Form $c_n(x)$4	Extract empirical convergence pattern
4	Split train/validation points	Enable GP fitting and diagnostics
5	Fit $c_n(x)$5	Learn smoothness and variance
6	Run Mahalanobis, PC, and coverage diagnostics	Test model adequacy
7	Infer $c_n(x)$6	Calibrate physical scales
8	Compute predictive truncation bands	Propagate correlated uncertainty

This workflow is reproducible through publicly available Jupyter notebooks, a point emphasized in both the 2024 assessment paper and the 2025 multi-potential study (Millican et al., 2024, Millican et al., 25 Aug 2025).

4. Convergence patterns, stationarity, and model adequacy

A major theme of the BUQEYE program is that the GP prior is not automatically valid for every EFT interaction or every parametrization. The 2024 assessment of semi-local momentum-space nucleon-nucleon potentials concluded that the BUQEYE model is “generally applicable” to the potential investigated, but also found that the assumption of GP stationarity across lab energy and scattering angle is “not generally met,” requiring adjustments in future work (Millican et al., 2024).

The later multi-potential study sharpened this point. It found that some “soft” regulator choices exhibit irregular convergence, specifically a mismatch between the sizes of coefficients at even and odd orders (Millican et al., 25 Aug 2025). In that setting, a simple stationary GP prior across $c_n(x)$7 is inadequate. The same paper reported that the GP length scale in the angular dimension, $c_n(x)$8, is approximately inversely proportional to the relative momentum, so the correlation structure is nonstationary in angle even when momentum smoothness is roughly stationary (Millican et al., 25 Aug 2025).

The proposed remediation is either a momentum-dependent warp of the angular coordinate,

$c_n(x)$9

or a nonstationary kernel in which $Q(x)$0 becomes a function of $Q(x)$1 (Millican et al., 25 Aug 2025). After this modification, diagnostics show significant improvement, and the BUQEYE model is validated for the retained hard potentials (Millican et al., 25 Aug 2025).

The literature therefore treats BUQEYE not as a fixed black-box prior, but as a statistically testable model class. Good practice begins by plotting the coefficient functions themselves to inspect whether they have roughly the same amplitude and wiggliness across orders, which the 2024 paper describes as naturalness and stationarity (Millican et al., 2024). A plausible implication is that BUQEYE’s strongest contributions are methodological as much as inferential: it supplies an explicit criterion for when EFT convergence patterns can be regarded as probabilistically regular and when they cannot.

5. Applications to nucleon-nucleon observables and Wolfenstein amplitudes

The BUQEYE model has been applied to multiple nucleon-nucleon datasets and observable classes. The 2024 assessment paper considered the semi-local momentum-space implementation of the chiral EFT expansion of the nucleon-nucleon potential and concluded that the model enables statistically principled estimates of the impact of higher EFT orders on observables (Millican et al., 2024). It also demonstrated posterior inference for the expansion parameter $Q(x)$2, the breakdown scale $Q(x)$3, the soft scale $Q(x)$4, and GP hyperparameters (Millican et al., 2024).

The 2025 Wolfenstein-amplitude analysis extends the same GP machinery to the five independent operator coefficients of the on-shell nucleon-nucleon amplitude. There, all five amplitudes $Q(x)$5, $Q(x)$6, $Q(x)$7, $Q(x)$8, and $Q(x)$9, with real and imaginary parts treated separately, are modeled by separate scalar GPs; there is no multi-output covariance in that work (McClung et al., 16 Jan 2025). At laboratory energies $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$0 MeV, one-dimensional GPs are built in the input variable $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$1, using training and testing point sets across the $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$2-range (McClung et al., 16 Jan 2025).

The study reports that, with the inferred breakdown scales, the EFT truncation uncertainties cover both higher-order results and empirical Wolfenstein amplitudes well for all orders other than the leading order (McClung et al., 16 Jan 2025). It further states that typical MAP breakdown scales are $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$3 MeV and typical correlation lengths are $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$4 MeV (McClung et al., 16 Jan 2025).

The later cross-potential convergence study broadens the application set to six common nucleon-nucleon scattering observables as functions of relative momentum and scattering angle (Millican et al., 25 Aug 2025). After excluding soft regulators with severe rms splitting and applying angular warping or equivalent nonstationary modeling, diagnostics pass for the remaining six potentials: SMS 450, 500, 550 MeV; SCS 0.9, 1.0 fm; and EMN 500 MeV (Millican et al., 25 Aug 2025).

6. Hyperparameter inference, breakdown scales, and broader BUQEYE methods

A central scientific use of the BUQEYE model is the inference of EFT scales from convergence data. Because the extracted coefficients depend on the choice of $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$5, and $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$6 depends on $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$7, these physical quantities become inferential targets rather than fixed external inputs (Millican et al., 2024). The 2024 paper recommends computing a joint likelihood over $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$8, combining it with priors, and marginalizing over $$\delta y_k(x)=y_{\rm ref}(x)\sum_{n=k+1}^\infty c_n(x)\,Q^n(x),$$9 to obtain $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$0 (Millican et al., 2024).

In the multi-potential analysis, with $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$1 fixed at the physical pion mass, the preferred breakdown scales for various N$c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$2LO interactions lie in the range $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$3--$c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$4 MeV (Millican et al., 25 Aug 2025). The same study reports that statistically consistent distributions for $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$5 across orders are only found for the SMS potential at a regulator scale of 450 or 500 MeV (Millican et al., 25 Aug 2025). When $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$6 and $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$7 are inferred simultaneously, joint MAPs lie in $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$8 MeV and $c_n(x)\overset{\rm iid}{\sim}\GP\bigl(\mu(x)=0,\;\sigma^2\,r(x,x';L)\bigr).$9 MeV depending on the potential, but the authors caution that posterior ellipses do not collapse as expected, indicating an identifiability tension between the two parameters (Millican et al., 25 Aug 2025).

The BUQEYE collaboration’s program also extends beyond truncation errors. Its guide to projection-based emulators defines emulators as fast surrogate models that replace expensive, high-fidelity nuclear-physics calculations by solving a much smaller reduced problem built from representative snapshots (Drischler et al., 2022). The reduction is based on variational and Galerkin projection methods together with offline–online decomposition. For bound states, inserting $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$0 into the Rayleigh–Ritz functional yields the reduced generalized eigenproblem

$$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$1

or, after orthonormalization, $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$2 with $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$3 (Drischler et al., 2022). This broader BUQEYE activity is methodologically adjacent to the truncation-error model: both are aimed at making Bayesian uncertainty quantification operational in computational nuclear theory.

7. Significance, limitations, and recurring misconceptions

The principal significance of the BUQEYE model is that it converts EFT truncation uncertainty from an informal order-estimate into a statistically calibrated predictive distribution with explicit correlation structure across kinematic variables (Millican et al., 2024). Because the same GP prior governs all orders, lower-order information constrains omitted higher-order contributions, and credible intervals can be checked against withheld calculations through Mahalanobis, pivoted-Cholesky, and coverage diagnostics (Millican et al., 2024, Millican et al., 25 Aug 2025).

A common misconception is that BUQEYE assumes all EFT calculations automatically satisfy a universal stationary GP prior. The published assessments argue otherwise. They show that the model’s success depends on a well-chosen expansion parameter $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$4, on regulator choices that produce regular convergence patterns, and on stationarity assumptions that may need to be relaxed or remapped, especially in the angular variable (Millican et al., 2024, Millican et al., 25 Aug 2025). Another misconception is that the breakdown scale is simply fixed once and for all. In the BUQEYE framework, $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$5 is often treated as a hyperparameter to be inferred from the coefficient data, with order dependence and posterior stability used as diagnostics of model adequacy (McClung et al., 16 Jan 2025, Millican et al., 25 Aug 2025).

The available literature also delineates clear limitations. Leading-order uncertainty bands can be too small in some channels, as seen for forward-angle tensor and spin-orbit Wolfenstein amplitudes (McClung et al., 16 Jan 2025). Soft regulator scales can generate even–odd coefficient splitting severe enough that the single-GP prior should not be used without modification (Millican et al., 25 Aug 2025). And simultaneous inference of $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$6 and $$K(x,x')=\sigma^2\exp\!\bigl[-\tfrac12\,(x-x')^T\,L^{-1}\,(x-x')\bigr],$$7 can exhibit identifiability problems (Millican et al., 25 Aug 2025).

Within those limits, BUQEYE provides a unified statistical language for EFT convergence assessment, correlated truncation-error propagation, and, through related projection-based emulators, fast many-query computation in nuclear theory (Drischler et al., 2022). This suggests that its lasting contribution is not merely a set of uncertainty bands for specific observables, but an integrated methodology in which EFT expansion structure, Bayesian inference, and model diagnostics are treated as parts of the same quantitative framework.