  An index of uniformity is developed as an alternative to the maximum-entropy
principle for selecting continuous, differentiable probability distributions
$\mathcal{P}$ subject to constraints $C$. The uniformity index developed in
this paper is motivated by the observation that among all differentiable
probability distributions defined on a finite interval $[a,b] \in \mathbb{R}$,
it is the uniform probability distribution that minimizes the path length of
the associated cumulative distribution function $F_{\mathcal{P}}$ on $[a,b]$.
This intuition is extended to situations where there are constraints on the
allowable probability distributions. In particular, constraints on the first
and second raw moments of a distribution are discussed in detail, including the
analytical form of the solutions and numerical studies of particular examples.
The resulting "shortest path" distributions are found to be decidedly more
heavy-tailed than the associated maximum-entropy distributions, suggesting that
entropy and "CDF path length" measure two different aspects of uncertainty for
bounded distributions.
