  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.
