Probabilistic definition of the perturbative theoretical uncertainty from missing higher orders

We consider the problem of quantifying the uncertainty on theoretical predictions based on perturbation theory due to missing higher orders. The most widely used approach, scale variation, is largely arbitrary and it has no probabilistic foundation, making it not suitable for robust data analysis. In 2011, Cacciari and Houdeau proposed a model based on a Bayesian approach to provide a probabilistic definition of the theory uncertainty from missing higher orders. In this work, we propose an improved version of the Cacciari-Houdeau model, that overcomes some limitations. In particular, it performs much better in case of perturbative expansions with large high-order contributions (as it often happens in QCD). In addition, we propose an alternative model based on the same idea of scale variation, which overcomes some of the shortcomings of the canonical approach, on top of providing a probabilistically-sound result. Moreover, we address the problem of the dependence of theoretical predictions on unphysical scales (such as the renormalization scale), and propose a solution to obtain a scale-independent result within the probabilistic framework. We validate these methods on expansions with known sums, and apply them to a number of physical observables in particle physics. We also investigate some variations, improvements and combinations of the models. We believe that these methods provide a powerful tool to reliably estimate theory uncertainty from missing higher orders that can be used in any physics analysis. The results of this work are easily accessible through a public code named THunc.