• Research Program on the Mathematics of Biodiversity, June-July 2012, Centre de Recerca Matemàtica, Barcelona, Spain. Organized by Ben Allen, Silvia Cuadrado, Tom Leinster, Richard Reeve and John Woolliams.

We’re having daily informal talks and there’s no way I can blog about all of them, talk to people here, and still get enough work done. So, I’ll just mention a few things that strike me! For example, this morning Lou Jost told me about an interesting paper by I. J. Good.

I’d known of I. J. Good as one of the guys who came up with the concept of a ‘technological singularity’. In 1963 he wrote:

Let an ultraintelligent machine be defined as a machine that can far surpass all the intellectual activities of any man however clever. Since the design of machines is one of these intellectual activities, an ultraintelligent machine could design even better machines; there would then unquestionably be an ‘intelligence explosion,’ and the intelligence of man would be left far behind. Thus the first ultraintelligent machine is the last invention that man need ever make.

He was a British mathematician who worked as a cryptologist at Bletchley Park with Alan Turing. After World War II, he continued to work with Turing on the design of computers and Bayesian statistics at the University of Manchester. Later he moved to the US. In 1968, thanks to his interest in artificial intelligence, he served as consultant for Stanley Kubrick’s film 2001: A Space Odyssey. He died in 2009.

Good was also a big chess enthusiast, and worked on writing programs to play chess. He’s the guy in front here:

If you want to learn more about his work on chess, click on this photo!

But the paper Lou Jost mentioned is on a rather different subject:

• Irving John Good, The population frequency of species and the estimation of population parameters, Biometrika 40 (1953), 237–264.

Let me just state one result, sloppily, without any details or precise hypotheses!

Puzzle: Suppose you go into the jungles of Ecuador and start collecting orchids. You count the number of orchids of each different species that you find. You get a list of numbers, something like this:

What is the chance that the next orchid you find will belong to a new species?

Good gives a rule of thumb for solving problems of this type:

Here is the total number of orchid you collected, and is the number of species for which you found exactly orchids of that species. In our example,

since we found just one orchid of three different species: those are the three 1’s at the end of our list. Furthermore,

So here is Good’s estimate the chance that the next orchid you collect will be of a new species:

Good’s argument is nontrivial—and of course it depends on some assumptions on the nature of the distribution of populations of different species! Since he doesn’t state these assumptions succinctly and I haven’t read the paper carefully yet, I’m afraid you’ll have to read the paper to find out what they are.

Of course the math works for samples of anything that comes in distinct types, not just species of organisms! Good considers four examples:

• moths captured in a light-trap at Rothamsted, England,

• words in American newspapers,

• nouns in Macaulay’s essay on Bacon,

• chess openings in games published by the British Chess Magazine in 1951.

By comparing a small sample to a bigger one, he studies how well his rule works in practice, and apparently it does okay.

In his paper, I. J. Good thanks Alan Turing for coming up with the basic idea. In fact he says Turing gave an ‘intuitive demonstration’ of it—but he doesn’t give this intuitive demonstration, and according to Lou Jost he actually admits somewhere that he forgot it.

You can read more about the idea here:

• Good–Turing frequency estimation.

By the way, Lou Jost is not only an expert on biodiversity and its relation to entropy! He lives in the jungles of Ecuador and has discovered over 60 new species of orchids, including the world’s smallest:

He found it in Ecuador, and the petals are just a few cells thick! (Typically, the news reports say he found it in Bolivia and the petals are just one cell thick.)

He said:

I found it among the roots of another plant that I had collected, another small orchid which I took back to grow in my greenhouse to get it to flower. A few months later I saw that down among the roots was a tiny little plant that I realised was more interesting than the bigger orchid. Looking at the flower is often the best way to be able to identify which species of orchid you’ve got hold of – and can tell you whether you’re looking at an unknown species or not.