  The combinatorial basis of entropy, given by Boltzmann, can be written $H =
N^{-1} \ln \mathbb{W}$, where $H$ is the dimensionless entropy, $N$ is the
number of entities and $\mathbb{W}$ is number of ways in which a given
realization of a system can occur (its statistical weight). This can be
broadened to give generalized combinatorial (or probabilistic) definitions of
entropy and cross-entropy: $H=\kappa (\phi(\mathbb{W}) +C)$ and $D=-\kappa
(\phi(\mathbb{P}) +C)$, where $\mathbb{P}$ is the probability of a given
realization, $\phi$ is a convenient transformation function, $\kappa$ is a
scaling parameter and $C$ an arbitrary constant. If $\mathbb{W}$ or
$\mathbb{P}$ satisfy the multinomial weight or distribution, then using
$\phi(\cdot)=\ln(\cdot)$ and $\kappa=N^{-1}$, $H$ and $D$ asymptotically
converge to the Shannon and Kullback-Leibler functions. In general, however,
$\mathbb{W}$ or $\mathbb{P}$ need not be multinomial, nor may they approach an
asymptotic limit. In such cases, the entropy or cross-entropy function can be
{\it defined} so that its extremization ("MaxEnt'' or "MinXEnt"), subject to
the constraints, gives the ``most probable'' (``MaxProb'') realization of the
system. This gives a probabilistic basis for MaxEnt and MinXEnt, independent of
any information-theoretic justification.
  This work examines the origins of the governing distribution $\mathbb{P}$....
(truncated)
