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

Minimizing the CDF Path Length: A Novel Perspective on Uniformity and Uncertainty of Bounded Distributions