  In this work we use Bayesian inference to quantitatively reconstruct the
solar properties most relevant to the solar composition problem using as inputs
the information provided by helioseismic and solar neutrino data. In
particular, we use a Gaussian process to model the functional shape of the
opacity uncertainty to gain flexibility and become as free as possible from
prejudice in this regard. With these tools we first readdress the statistical
significance of the solar composition problem. Furthermore, starting from a
composition unbiased set of standard solar models we are able to statistically
select those with solar chemical composition and other solar inputs which
better describe the helioseismic and neutrino observations. In particular, we
are able to reconstruct the solar opacity profile in a data driven fashion,
independently of any reference opacity tables, obtaining a 4% uncertainty at
the base of the convective envelope and 0.8% at the solar core. When systematic
uncertainties are included, results are 7.5% and 2% respectively. In addition
we find that the values of most of the other inputs of the standard solar
models required to better describe the helioseismic and neutrino data are in
good agreement with those adopted as the standard priors, with the exception of
the astrophysical factor $S_{11}$ and the microscopic diffusion rates, for
which data suggests a 1% and 30% reduction respectively. As an output of the
study we derive the corresponding data driven predictions for the solar
neutrino fluxes.
