  Some recent investigations of spectropolarimetric observations of the Zeeman
effect in the Fe I lines at 630 nm carried out with the Hinode solar space
telescope have concluded that the strength of the magnetic field vector in the
internetwork regions of the quiet Sun is in the hG regime and that its
inclination is predominantly horizontal. We critically reconsider the analysis
of such observations and carry out a complete Bayesian analysis with the aim of
extracting as much information as possible from them, including error bars. We
apply the recently developed BayesME code that carries out a complete Bayesian
inference for Milne-Eddington atmospheres. The sampling of the posterior
distribution function is obtained with a Markov Chain Monte Carlo scheme and
the marginal distributions are analyzed in detail. The Kullback-Leibler
divergence is used to study the extent to which the observations introduce new
information in the inference process resulting in sufficiently constrained
parameters. Our analysis clearly shows that only upper limits to the magnetic
field strength can be inferred with fields in the kG regime completely
discarded. Furthermore, the noise level present in the analyzed Hinode
observations induces a substantial loss of information for constraining the
azimuth of the magnetic field. Concerning the inclination of the field, we
demonstrate that some information is available to constrain it for those pixels
with the largest polarimetric signal. The results also point out that the field
in pixels with small polarimetric signals can be nicely reproduced in terms of
a quasi-isotropic distribution.
