Mathematical diversity of parts for a continuous distribution

The current paper is part of a series exploring how to link diversity measures (e.g., Gini-Simpson index, Shannon entropy, Hill numbers) to a distribution's original shape and to compare parts of a distribution, in terms of diversity, with the whole. This linkage is crucial to understanding the exact relationship between the density of an original probability distribution, denoted by $p(x)$, and the diversity $D$ in non-uniform distributions, both within parts of a distribution and the whole. This linkage is empirically useful because most real-world systems have unequal distributions and consist of multiple diversity types with unknown and changing frequencies at different levels of scale (e.g., income diversity, economic complexity indices, rankings). To date, we have proven our results for discrete distributions. Our focus here is continuous distributions. In both instances, we do so by linking case-based entropy, a diversity approach we developed, to a probability distribution’s shape for continuous distributions. This allows us to demonstrate that the original probability distribution $g_1$, the case-based entropy curve $g_2$, and the slope of diversity $g_3$ ($c_{(a,x)}$ versus the $c_{(a,x)}* \ln A_{(a,x)}$ curve) are one-to-one (or injective). In other words, a different probability distribution $g_1$, results in different curves for $g_2$, and $g_3$. Therefore, a different permutation of the original probability distribution (resulting in a different shape) will uniquely determine the graphs $g_2$ and $g_3$. By proving our approach’s injective nature for continuous distributions, we establish a unique method to measure the degree of uniformity as measured by $D/c$ and show a unique way to compute $D/c$ for various shapes of the original continuous distribution to compare different distributions and their parts.