Multi-variate joint PDF for non-Gaussianities: exact formulation and
  generic approximations
  We provide an exact expression for the multi-variate joint probability
distribution function of non-Gaussian fields primordially arising from local
transformations of a Gaussian field. This kind of non-Gaussianity is generated
in many models of inflation. We apply our expression to the non- Gaussianity
estimation from Cosmic Microwave Background maps and the halo mass function
where we obtain analytical expressions. We also provide analytic approximations
and their range of validity. For the Cosmic Microwave Background we give a fast
way to compute the PDF which is valid up to 7{\sigma} for fNL values (both true
and sampled) not ruled out by current observations, which consists of
expressing the PDF as a combination of bispectrum and trispectrum of the
temperature maps. The resulting expression is valid for any kind of
non-Gaussianity and is not limited to the local type. The above results may
serve as the basis for a fully Bayesian analysis of the non-Gaussianity
parameter.
