  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
$P(\mathrm{parametrization}\,|\,\mathrm{data})$ 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 turns out to be
important to obtain the agreement between the $r_E^p$ and $r_M^p$ values. The
results disagree with recent muonic atom measurements.
