A class of parametric distribution functions was proposed in (Di Troia 2012 Plasma Phys. Control. Fusion54 105017) as equilibrium distribution functions (EDFs) for charged particles in fusion plasmas, representing supra-thermal particles in anisotropic equilibria for Neutral Beam Injection and Ion Cyclotron Heating scenarios. Moreover, those EDFs can be used to represent also nearly isotropic equilibria for Slowing-Down alpha particles and core thermal plasma populations. Such EDFs depend on constants of motion (COMs). In axisymmetric system with no equilibrium electric field, they depend on toroidal canonical momentum , kinetic energy w and magnetic moment μ.In the present work, the same EDFs are obtained from first principles and general hypothesis. The derivation is probabilistic and makes use of the Bayes' Theorem. The bayesian argument is used to describe how the plasma is far from the prior probability distribution function (pdf), e.g. Maxwellian, based on the information obtained from magnetic moment and guiding center velocity pdf.Once the general functional form of the EDF has been settled, it is shown how to associate a modified Landau collision operator in the Fokker–Planck equation, to describe the system relaxation towards the proposed EDF.