• S. Chakravarty and W. Eichhorn, An axiomatic characterization of a generalized entropy index of concentration, Journal of Productivity Analysis 2 (1991) 103–112.

• Heikki Pursiainen, Consistent Aggregation Methods and Index Number Theory, Ph.D Thesis, Faculty of Social Sciences, University of Helskini, 2005.

* http://www.anc.upmc.fr/biblioNca/pdf/Brasselet2011a.pdf

(the interesting part is between p.7 and p.14.)

It only deals with mutual information between 2 sets of which only one is equipped with a distance. That’s why there is a caveat in the additivity property. The version with 2 sets equipped with a distance is more elegant but yet to be published.

The gap between communication theory and biodiversity is rather wide and I don’t claim that it can be useful here. But it’s interesting to see that similar ideas appeared in so different fields.

• V. Stevens, S. Pavoine and M. Baguette, Variation within and between closely related species uncovers high intra-specific variability in dispersal, PloS ONE (2010) 5:e11123.

Just thinking out loud here, but wouldn’t mutual information rather than entropy be helpful in such cases?
If we consider a set of species and a set of individuals, then we can define entropies , and also on the set of individuals of a single species . These different measures could account for the biodiversity of species, across species and within species. Though I’m not sure exactly how to interpret it in terms of biodiversity, we could even compute an information between these two sets.

Funnily enough, in 2009, in a different context (computational neuroscience), I defined an entropy on sets equipped with a distance that is identical to that of Tom Leinster (or Ricotta and Szeidl 2006 which I did not know at that time). But instead of using Renyi entropy, I sticked to Shannon and studied the mutual information (MI) on sets equipped with distances.

In biodiversity, the MI would tell you about how species are different compared to their inner variability. Or conversely, their inner variability would appear as a reference “scale” to measure the between-species variability.

But again, just thinking out loud here (and incidentally indecently advertised my work…)

It seems that species are always taken as predefined and that all individuals inside a species are considered identical. So that the variability “within” species is not taken into account.

There’s nothing in the underlying mathematics that forces us to take this attitude: I’m using the word ‘species’ for an element of the set on which our probability distribution is given, but it doesn’t need to be a set of biological species in the traditional sense.

Even better, we could consider a set equipped with a distance function, so that individuals contribute to the biodiversity more when they are more dissimilar from other individuals that are present. This would answer your objection, at least in theory. Incorporating a distance function requires new mathematical ideas—but luckily, these have been developed very nicely here:

• Christina Cobbold and Tom Leinster, Measuring diversity: the importance of species similarity, Ecology 93 (2012), 477–489.

The title says ‘species similarity’ but mathematically we could just as well talk about the similarity of one individual to another, and how this affects biodiversity.

If the paper looks too hard to read, start with Tom’s post on this blog!

In practice, however, you are right. Researchers have found it difficult to get enough data about variability within species to include this effect when measuring biodiversity. I’ve heard people at this conference complaining about this. There are a lot of interesting issues—mathematical, statistical and experimental—involved in trying to remedy this problem!