Hello! The people attending the Mathematics of Biodiversity conference are very interested in two other conditions that a diversity index should satisfy, namely that it be an ‘effective number’ and that it obey the ‘replication principle’. I explained these conditions in Part 4. The Hill numbers are favored as diversity indices because they obey these conditions (as well as the ones you list), while the Shannon and Rényi entropies do not. In his talk, Lou Jost explained some serious mistakes that people have made by working with diversity indices that do not obey these additional conditions. I also recommend these:

• Lou Jost, Entropy and diversity, *Oikos* **113** (2006), 363–375.

• Tom Leinster, Measuring biodiversity, *Azimuth*, 7 November 2011.

Thanks for pointing out your papers! I’ll tell Tom and Lou about them.

]]>1. Zhang, Z. and Zhou, J. (2010). Re-parameterization of multinomial distributions and diversity indices. Journal of Statistical Planning and Inference 140, pp. 1731–1738, 2010;

2. Zhang, Z. (2012). Entropy estimation in Turing’s perspective. Neural Computation, 24, pp. 1368—1389.

I have been thinking about the question what constitutes a biodiversity index for years and the following is a summary of what I believe currently:

1. The word “biodiversity” clearly conveys certain intuitive meaning. However people seem very hesitant to give mathematical definitions for diversity. This puzzles me. The only reason I could see is that “diversity” may be reasonably understood in two different ways, one involving the total number of species (K) and another involving the species proportions (p_{k}). Personally I believe the latter is the real issue – not that K is not important, it is and it is well defined.

2. The lack of (universally accepted) definition of diversity hinders the methodological advancement in issues such as statistical estimation. We need to develop (good) general definitions of diversity.

Pertaining to 2 above, I wish to see some discussion or comments on the minimal set of conditions a “diversity index” must satisfy. I can only think of three:

1. It must be non-negative (this could be a superficial one).

2. It must be permutation (with respect to the letters of the alphabet) invariant.

3. It attains its minimum (e.g. 0) when p_{k}=1 for some k.

Can you think of any other? Maybe this could be the beginning of a fruitful discussion.

]]>Romain has recently edited this post to include more information, and that reference is now among those he lists!

]]>Hmm, that’s not it, but it looks interesting.

]]>This whole series of posts is about the Research Program on the Mathematics Biodiversity, at CRM, which lasts from June 18 to July 20, 2012. I edited this post to clarify that this is the ‘program’ I was talking about.

]]>I. Nemenman, F. Shafee, and W. Bialek, “Entropy and inference, revisited,” Advances in neural information processing, vol. 14, 2002.

See also:

http://www.menem.com/~ilya/wiki/index.php/Entropy_Estimation

Perhaps this is it http://fens2012.neurosciences.asso.fr/

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