  We present an approximate calculation of the full Bayesian posterior
probability distribution for the local non-Gaussianity parameter
$f_{\text{nl}}$ from observations of cosmic microwave background anisotropies
within the framework of information field theory. The approximation that we
introduce allows us to dispense with numerically expensive sampling techniques.
We use a novel posterior validation method (DIP test) in cosmology to test the
precision of our method. It transfers inaccuracies of the calculated posterior
into deviations from a uniform distribution for a specially constructed test
quantity. For this procedure we study toy cases that use one- and
two-dimensional flat skies, as well as the full spherical sky. We find that we
are able to calculate the posterior precisely under a flat-sky approximation,
albeit not in the spherical case. We argue that this is most likely due to an
insufficient precision of the used numerical implementation of the spherical
harmonic transform, which might affect other non-Gaussianity estimators as
well. Furthermore, we present how a nonlinear reconstruction of the primordial
gravitational potential on the full spherical sky can be obtained in principle.
Using the flat-sky approximation, we find deviations for the posterior of
$f_{\text{nl}}$ from a Gaussian shape that become more significant for larger
values of the underlying true $f_{\text{nl}}$. We also perform a comparison to
the well-known estimator of Komatsu et al. [Astrophys. J. 634, 14 (2005)] and
finally derive the posterior for the local non-Gaussianity parameter
$g_{\text{nl}}$ as an example of how to extend the introduced formalism to
higher orders of non-Gaussianity.
