My Entropy ‘Pearl’: Using Turing’s Insight to Find an Optimal Estimator for Shannon Entropy

Post provided by Anne Chao (National Tsing Hua University, Taiwan)

Shannon Entropy

Not quite as precious as my entropy pearl

Not quite as precious as my entropy pearl ©Amboo Who

Ludwig Boltzmann (1844-1906) introduced the modern formula for entropy in statistical mechanics in 1870s. Since its generalization by Claude E. Shannon in his pioneering 1948 paper A Mathematical Theory of Communication, this entropy became known as ‘Shannon entropy’.

Shannon entropy and its exponential have been extensively used to characterize uncertainty, diversity and information-related quantities in ecology, genetics, information theory, computer science and many other fields. Its mathematical expression is given in the figure below.

In the 1950s Shannon entropy was adopted by ecologists as a diversity measure. It’s interpreted as a measure of the uncertainty in the species identity of an individual randomly selected from a community. A higher degree of uncertainty means greater diversity in the community.

Unlike species richness which gives equal weight to all species, or the Gini-Simpson index that gives more weight to individuals of abundant species, Shannon entropy and its exponential (“the effective number of common species” or diversity of order one) are the only standard frequency-sensitive complexity measures that weigh species in proportion to their population abundances. To put it simply: it treats all individuals equally. This is the most natural weighing for many applications. Continue reading