Shannon diversity index

Suppose that $R(\pi_k) = -\ln(\pi_k)$. This corresponds, roughly, to saying that a species that is rarely encountered is almost infinitely rare while a species that is commonly encountered is not rare at all. This might be appropriate if we think of $R(\pi_k)$ as measuring how much value we place on a species as a function of its frequency of occurrence in a community. Then

$$\begin{eqnarray*} \Delta_0(C) &=& \sum_{k=1}^s\pi_k\left[-\ln(\pi_k)\right] \cr &=& -\sum_{k=1}^s\pi_k\ln(\pi_k) \cr \end{eqnarray*}$$