[Apologies to those of you who may have seen an inadvertantly-published unfinished version of this post]
I’ve just returned from a week at the Annual meeting of the Institute for Mathematical Statistics in Gothenburg, Sweden. It’s always instructive to go to meetings outside of one’s specialty, outside of the proverbial comfort zone. I’ve been in my own field long enough that I’m used to feeling like one of the popular kids, knowing and being known by most of my fellow cosmologists — it’s a good corrective to an overinflated sense of self-worth to be somewhere where nobody knows your name. Having said that, I was bit disappointed in the turnout for our session, “Statistics, Physics and Astrophysics”. Mathematical statistics is a highly specialized field, but with five or more parallel sessions going on at once, most attendees could find something interesting. However, even cross-cutting sessions of supposedly general interest — our talks were by physicists, not statisticians — didn’t have the opportunity to get a wide audience.
The meeting itself, outside of that session, was very much of mathematical statistics, more about lemmas and proofs than practical data analysis. Of course these theoretical underpinnings are crucial to the eventual practical work, although it’s always disheartening to see the mathematicians idealise a problem all out of recognition. For example, the mathematicians routinely assume that the errors on a measurement are independent and identically distributed (“iid” for short) but in practice this is rarely true in the data that we gather. (I should use this as an opportunity to mention my favourite statistics terms of art: homoscedastic and heteroscedastic, describing, respectively, identical and varying distributions.)
But there were more than a couple of interesting talks and sessions, mostly concentrating upon two of the most exciting — and newsworthy — intersections between statistical problems and the real world: finance and climate. How do we compare complicated, badly-sampled, real-world economic or climate data to complicated models which don’t pretend to capture the full range of phenomena? In what sense are the predictions inherently statistical and in what sense are they deterministic? “Probability”, said de Finetti, the famous Bayesian statistician “does not exist”, by which he meant that probabilities are statements about our knowledge of the world, not statements about the world. The world does, however, give sequences of values (stock prices, temperatures, etc.) which we can test our judgements against. This, in the financial realm, was the discussion of Hans Föllmer’s Medallion Prize Lecture, which veered into the more abstract realm of stochastic integration, Martingales and Itō calculus along the way.
Another pleasure was the session chaired by Robert Adler. Adler is the author of a book called The Geometry of Random Fields, a book which has had a significant effect upon cosmology from the 1980s through today. A “random field” is something that you could measure over some regime of space and time, but for which your theory doesn’t determine its actual value, but only its statistical properties, such as its average and the way the value at different points are related to one another. The best example in cosmology is the CMB itself — none of our theories predict the temperature at any particular place, but the theories that have survived our tests make predictions about the mean value and about the product of temperatures at any two points — this is called the correlation function, and a random field in which only the mean and correlation function can be specified is called a Gaussian random field, after the Gaussian distribution that is the mathematical version of this description. Indeed, Adler uses the CMB as one of the examples on his academic home page. But there are many more application besides: the session featured talks on brain imaging and on Google’s use random fields to analyze data about the way people look at their web pages
Gothenburg itself was nice in that Scandinavian way: nice, but not terribly exciting, full of healthy, attractive people who seem pleased with their lot in life. The week of our meeting overlapped with two other important events in the town. The other big meeting in town was the World Library and Information Congress — you can only imagine the party atmosphere in a town filled with both statisticians and librarians! But adding to that, Gothenburg was hosting its summer kulturkalas festival of culture — the streets were filled with musicians and other performers to distract us from the mathematics.