Proton radius from Bayesian inference

The methods of Bayesian statistics are used to extract the value of the proton radius from the elastic $ep$ scattering data in a model-independent way. To achieve that goal a large number of parametrizations (equivalent to neural network schemes) are considered and ranked by their conditional probability $P\left(\mathrm{parametrization}\right|\mathrm{data})$ instead of using the minimal error criterion. As a result the most probable proton radii values $(r_E^p=0.899\pm 0.003$ fm, $r_M^p=0.879\pm 0.007$ fm) are obtained and systematic error due to freedom in the choice of parametrization is estimated. Correcting the data for the two-photon-exchange effect leads to smaller differences between the extracted values of $r_E^p$ and $r_M^p$. The results disagree with recent muonic atom measurements.

Figure 1

Scheme of the neural network (with $H=4$ hidden units) used in the analysis, for reference see Eq. (6). Every line (connection) corresponds to one parameter $w_i$. Gray circles denote the sigmoid functions, open circles are constants.

Figure 2

Logarithm of evidence for the best neural networks in each scheme. Solid (dashed) lines correspond to the analyses of the data corrected (not corrected) by the TPE effect. Points mark the best models for each analysis, $\square$ for the cutoff at $Q^2<1$ ${\mathrm{GeV}}^2,◯$ for $Q^2<3$ ${\mathrm{GeV}}^2$, and $▵$ for $Q^2<10$ ${\mathrm{GeV}}^2$.

Figure 3

Form factor $G_E/G_D$ $\left[G_D=1/{(1+Q^2/0.71{\mathrm{GeV}}^2)}^2\right]$. Solid, dotted, and dashed lines correspond to the best models in the analysis with data limited by $Q_{\mathrm{cutoff}}^2=1,3,$ and 10 ${\mathrm{GeV}}^2$ respectively. The three shaded areas at the bottom of the plot denote $1\sigma$ uncertainty due to the change in the fit parameters (calculated within the Hessian approximation). The bottom, middle, and top areas correspond to the 1, 3, 10 ${\mathrm{GeV}}^2$ cuttoff values respectively. The gray solid thin lines show models which are best for neural networks with definite number of hidden units, but are not globally best.

Figure 4

Proton radii corresponding to the models which are the best within particular data selection and neural network scheme. The results for the data corrected by the TPE are marked with black $▵,\square$, and $◯$ for $Q_{\mathrm{cutoff}}^2=1$, 3, and 10 ${\mathrm{GeV}}^2$ respectively. The results for the uncorrected data are marked with blue $\nabla$ and $\diamond$ for $Q_{\mathrm{cutoff}}^2=1$ and 3 ${\mathrm{GeV}}^2$ respectively. The filled points mark the best result for each case. The three leftmost red points are the best results according to the minimum of the error function. For clarity their x coordinate has been changed. Their ln(evidence) values are $-718,-1070$, and $-1311$. The picture is based on the analysis of $39\times 5=195$ models.