  A class of parametric distribution functions has been proposed in [C.DiTroia,
Plasma Physics and Controlled Fusion,54,2012] as equilibrium distribution
functions (EDFs) for charged particles in fusion plasmas, representing
supra-thermal particles in anisotropic equilibria for Neutral Beam Injection,
Ion Cyclotron Heating scenarios. Moreover, the EDFs can also represent nearly
isotropic equilibria for Slowing-Down $alpha$ particles and core thermal plasma
populations. These EDFs depend on constants of motion (COMs). Assuming an
axisymmetric system with no equilibrium electric field, the EDF depends on the
toroidal canonical momentum $P_\phi$, the kinetic energy $w$ and the magnetic
moment \mu. In the present work, the EDFs are obtained from first principles
and general hypothesis. The derivation is probabilistic and makes use of the
Bayes' Theorem.
  The bayesian argument allows us to describe how far from the prior
probability distribution function (pdf), e.g. Maxwellian, the plasma is, based
on the information obtained from magnetic moment and GC velocity pdf.
  Once the general functional form of the EDF has been settled, it is shown how
to associate a Landau collision operator and a Fokker-Planck equation that
ensures the system relaxation towards the proposed EDF.
