Bayesian analysis of systematic errors in the determination of the constant of gravitation

Measurements of the gravitational constant $G$ are notoriously difficult. Individual state-of-the-art experiments have managed to determine the value of $G$ with high precision: although, when considered collectively, the range in the measured values of $G$ far exceeds individual uncertainties, suggesting the presence of unaccounted for systematic effects. Here, we propose a Bayesian framework to account for the presence of systematic errors in the various measurement of $G$ while proposing a consensus value, following two paths: a parametric approach, based on the Maximum Entropy Principle, and a non-parametric one, the latter being a very flexible approach not committed to any specific functional form. With both our methods, we find that the uncertainty on this fundamental constant, once systematics are included, is significantly larger than what quoted in CODATA 2018. Moreover, the morphology of the non-parametric distribution hints towards the presence of several sources of unaccounted for systematics. In light of this, we recommend a consensus value for the gravitational constant $G = 6.6740^{+0.0015}_{-0.0015} \times 10^{-11}\ \mathrm{m}^3\ \mathrm{kg}^{-1}\ \mathrm{s}^{-2}$.