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

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.