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Quasi-Gaussian Missing Higher Order Uncertainties

This paper reviews Bayesian models for estimating uncertainties from missing higher-order perturbative predictions and demonstrates how modifying their likelihoods and priors can yield smooth, quasi-Gaussian probability distributions that better align with standard physics analysis assumptions.

Original authors: Marco Bonvini, Emanuele Bagnaschi, Lorenzo Paparella

Published 2026-09-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Marco Bonvini, Emanuele Bagnaschi, Lorenzo Paparella

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of high-energy physics, scientists act as cosmic accountants, trying to tally the precise energy and momentum of particles that collide at nearly the speed of light. To do this, they rely on mathematical predictions that describe how these particles behave. However, these calculations are never perfectly complete. Because the equations are incredibly complex, physicists must stop their calculations at a certain point, leaving behind a small, uncalculated remainder. This missing piece is a source of uncertainty, a shadow of doubt that hangs over every prediction. If a scientist wants to know if a new discovery is real or just a fluke, they must know exactly how big this shadow is. Traditionally, they have estimated this uncertainty by making a rough guess based on how much the result changes when they tweak a few numbers in their equations. This method works, but it treats the uncertainty as a simple, bell-shaped curve, a standard shape that assumes errors are distributed evenly around a central value.

A team of researchers has now revisited this problem, asking whether that standard assumption is actually correct. They explored a more sophisticated way of thinking about these missing pieces using a framework called Bayesian inference. Instead of just guessing, this approach treats the unknown parts of the calculation as hidden variables with their own rules. The researchers built a model that starts with what is known and uses probability to estimate what is unknown, generating a full map of possible outcomes rather than a single number. When they first applied this model, the results were surprising. The maps they produced did not look like the smooth, familiar bell curves physicists are used to. Instead, the shapes were jagged, with sharp spikes, flat plateaus, and uneven edges. While these shapes were mathematically honest reflections of the underlying rules, they were difficult to use in real-world analysis, where most tools are built to handle the smooth, standard bell curve.

The core of this new work was to see if the researchers could gently reshape these jagged maps into something that looked more like the familiar bell curve, without losing the honesty of the original model. They realized that the strange shapes came from the specific rules they had chosen for how the unknown parts behave. By slightly adjusting these rules—specifically how much weight is given to different possible values and how the boundaries are set—they could smooth out the jagged edges. They tested several different ways to soften the rules, trying shapes that favored values near the center and gently tapered off at the edges. The goal was not to force the data into a box, but to find a set of rules that naturally produced a smoother, more usable distribution while still respecting the fundamental physics.

The results of this experiment were mixed but promising. When they applied these new, smoothed-out rules to a specific type of particle collision involving the Higgs boson, the outcome depended on which version of the model they used. For one version, which relied on how the calculation changed with a specific scale, the new rules worked beautifully. The resulting distribution became almost indistinguishable from a standard bell curve, with a smooth peak and tails that faded away just as expected. This version also kept a crucial mathematical relationship between the width of the curve and the range of likely values, meaning physicists could trust the standard tools to analyze it. For another version of the model, the results were smoother than before and looked more like a bell curve, but they still retained some distinct features, such as a sharper peak and heavier tails. In this case, forcing the data into a standard bell shape would still miss some of the unique character of the result.

The researchers concluded that it is possible to create these "quasi-Gaussian" distributions that are both physically honest and practically useful, but it requires choosing the right set of rules for the job. They found that for models based on how the calculation changes with scale, the adjustments worked perfectly, allowing the complex, jagged reality of the missing pieces to be represented by a smooth, familiar shape. For more general models, the improvement was significant but not perfect; the shapes became much smoother, yet they still held onto some of their original, unique character. This work provides a new toolkit for physicists, offering a way to handle the uncertainty of missing calculations with greater precision and flexibility. It suggests that while the universe may not always follow the simplest, smoothest patterns, we can find mathematical ways to describe those patterns that are both accurate and easy to work with, bridging the gap between complex theory and practical discovery.

Technical Summary: Quasi-Gaussian Missing Higher Order Uncertainties

Problem Statement
Perturbative predictions in particle physics are inherently truncated, leaving an uncalculated remainder ( $\Delta_{\text{MHO}}$ ) that constitutes a source of theoretical uncertainty. While the standard practice involves varying renormalization and factorization scales to estimate this uncertainty, this approach relies on conventional choices regarding the variation range and central scale, lacking a rigorous probabilistic interpretation. Bayesian inference offers a framework to quantify this uncertainty by treating the behavior of the perturbative expansion as a set of hypotheses encoded in likelihoods and priors. However, existing Bayesian models (specifically those based on geometric bounds and scale dependence) often yield posterior probability distributions with non-Gaussian features, such as spikes, plateaus, and sharp edges. Since most phenomenological analyses assume Gaussian uncertainties, propagating these non-Gaussian distributions is non-trivial.

Methodology
The authors revisit Bayesian models for Missing Higher Order (MHO) uncertainties, specifically the geometric model and models based on scale dependence introduced in Ref. [3]. The generic perturbative series is factored as $\Sigma(\mu) = \Sigma_{\text{LO}}(\mu) \sum_{k \ge 0} \delta_k(\mu)$ , where $\delta_k$ are dimensionless coefficients.

The study focuses on modifying the likelihood functions and priors to induce posterior distributions that resemble Gaussian shapes, without altering the underlying physical constraints.

Likelihood Modifications: Instead of the standard flat likelihood (uniform distribution within a bound defined by parameters $c, a$ ), the authors introduce generalized likelihoods where the probability density function $f(x)$ favors smaller values of the coefficients within the allowed range. Three specific functional forms for $f(x)$ are tested:
- Parabolic: $f(x) = 1 - x^2$
- Cosine: $f(x) = \cos^2(\pi x/2)$ (providing a smooth transition to zero at the boundaries)
- Gaussian: $f(x) = \exp(-5x^2)$ (with truncated tails)
Prior Modifications: The authors target parameters with infinite ranges ( $c, \lambda, \eta$ ) which govern the tails of the posterior distributions. The default priors in previous works are power-law based, which do not yield Gaussian tails. To achieve exponential suppression, two variants are proposed:
- Exponential: $P_0(c) \propto c^\kappa \exp(-c)$
- Incomplete Gamma: $P_0(c) \propto \Gamma(s, c)$

Key Contributions and Results
The paper systematically tests combinations of these modified likelihoods and priors, applying them to the prototypical case of inclusive gluon-fusion Higgs production at the LHC (13 TeV, $m_H = 125$ GeV). The results are analyzed for the geometric, scale-variation, and constrained scale-variation models:

Geometric Model: Using the cosine likelihood and exponential prior produces smooth, bell-shaped distributions. However, a detailed inspection reveals that the proportions are not strictly Gaussian; the distributions exhibit taller peaks and higher tails than a true Gaussian.
Scale-Variation Model: The combination of the cosine likelihood and exponential prior yields distributions that are very similar to Gaussians. This is verified by the observation that the standard deviation is almost identical to the 68% credibility interval, and the 95% interval is approximately twice the 68% interval, consistent with Gaussian statistics.
Constrained Scale-Variation Model: This model produces smoother distributions than the default implementation but retains some spikes due to the non-linear nature of the constraint on scale dependence ( $r_k$ ). The resulting distributions are asymmetric. Nevertheless, statistical estimators approximately satisfy Gaussian relations, allowing for a faithful Gaussian approximation of the features.

Significance and Claims
The authors claim that by suitably modifying the likelihood and priors, it is possible to obtain "quasi-Gaussian" probability distributions for MHO uncertainties while retaining the explicit physical assumptions of the Bayesian framework.

Practical Utility: The primary significance is the facilitation of propagating theory uncertainties in phenomenological analyses that currently rely on Gaussian assumptions.
Model Specificity: The paper modestly notes that while the goal is achieved for models based on scale dependence (producing distributions that satisfy Gaussian statistical relations), the geometric model yields smoother, Gaussian-like shapes that do not perfectly match Gaussian proportions. In the latter case, a Gaussian approximation might miss qualitative features of the model.
Transparency: The approach maintains the advantage of Bayesian methods: all assumptions are explicit, testable, and modifiable, unlike the conventional scale-variation method.

The work concludes that for scale-variation-based models, a Gaussian approximation is a valid and faithful representation of the uncertainty, whereas for the general geometric model, the approximation is smoother but less precise in its statistical proportions.

Technical Summary: Quasi-Gaussian Missing Higher Order Uncertainties

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