  We revise the Bayesian inference steps required to analyse the cosmological
large-scale structure. Here we make special emphasis in the complications which
arise due to the non-Gaussian character of the galaxy and matter distribution.
In particular we investigate the advantages and limitations of the
Poisson-lognormal model and discuss how to extend this work. With the lognormal
prior using the Hamiltonian sampling technique and on scales of about 4 h^{-1}
Mpc we find that the over-dense regions are excellent reconstructed, however,
under-dense regions (void statistics) are quantitatively poorly recovered.
Contrary to the maximum a posteriori (MAP) solution which was shown to
over-estimate the density in the under-dense regions we obtain lower densities
than in N-body simulations. This is due to the fact that the MAP solution is
conservative whereas the full posterior yields samples which are consistent
with the prior statistics. The lognormal prior is not able to capture the full
non-linear regime at scales below ~ 10 h^{-1} Mpc for which higher order
correlations would be required to describe the matter statistics. However, we
confirm as it was recently shown in the context of Ly-alpha forest tomography
that the Poisson-lognormal model provides the correct two-point statistics (or
power-spectrum).
