Generalized classical, quantum and intermediate statistics and the Pólya urn model

Generalized probability distributions for Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics, with unequal source (“prior”) probabilities qiqi for each level i, are obtained by combinatorial reasoning. For equiprobable degenerate sublevels, these reduce to those given by Brillouin in 1930, more commonly given as a statistical weight for each statistic. These distributions and corresponding cross-entropy (divergence) functions are shown to be special cases of the Pólya urn model, involving neither independent nor identically distributed (“ninid”) sampling. The most probable Pólya distribution is shown to contain the Acharya–Swamy intermediate statistic.