A multiline Bayesian analysis of the Zeeman broadening in the solar atmosphere is presented. A hierarchical probabilistic model, based on the simple but realistic Milne-Eddington approximation to the solution of the radiative transfer equation is used to explain the data in the optical and near infrared. Our method makes use of the full line profiles of more than 500 spectral lines from 4000 Å to 1.8 μm. Although the problem suffers from a strong degeneracy between the magnetic broadening and any other remaining broadening mechanism, the hierarchical model allows the magnetic contribution to be isolated with reliability. We obtain the cumulative distribution function for the field strength and use it to put reliable upper limits on the unresolved magnetic field strength in the solar atmosphere. The field is below 160-180 G with a 90% probability.