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This article is a little different from the fare you’re used to getting from me: it’s for SIAM News, which is the glossy magazine for members of the Society for Industrial and Applied Mathematics. The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. In this case, I covered research by mathematicians looking at a type of system that occurs in biology and materials science. While the article contains equations, I wrote it to be understandable if you skim that part.
For SIAM News:
Self-organization is an important topic across scientific disciplines. Be it the spontaneous flocking of birds or dramatic phase transitions like superconductivity in materials, collective behavior without underlying intelligence occurs everywhere.
Many of these behaviors involve synchronization, or self-organization in time, such as activation in heart cells or the simultaneous blinking of certain firefly species. Others are aggregations, or self-organization in space, like swarming insects, flocking birds, or the alignment of electron spins in magnetic material.
Despite their conceptual similarity, self-organization in space and time have largely been treated separately. “I was curious about whether the two fields had been wedded, and it turns out they hadn’t, at least not fully,” Kevin O’Keeffe, a postdoctoral researcher at the Massachusetts Institute of Technology, said. “I knew all these tricks and mathematical tools from synchronization, and I was looking to cross-fertilize them into the swarming world.”
[Read the rest at SIAM News…]
Statistics is rarely sexy, sometimes satisfying, occasionally misused, but useful enough that more people should know how to use it than do. (Insert obvious condom joke here.) However, a particular method in statistics got additional attention last fall during the United States national elections: Bayesian inference. I wrote two pieces last week, drawing from a recent Science article, that highlighted Bayesian methods. The first (written for Ars Technica, now picked up at WiredUK!) was about Bayes’ theorem and why its use is still controversial for some people.
Bayes’ theorem in essence states that the probability of a given hypothesis depends both on the current data and prior knowledge. In the case of the 2012 United States election, Silver used successive polls from various sources as priors to refine his probability estimates. (In other words, saying he “predicted” the outcome of the election is slightly misleading: he calculated which candidate was most likely to win in each state based on the polling data.) In other cases, priors could be the outcome of earlier experiments or even educated assumptions drawn from experience. The wise statistician or scientist constructs priors that are informative, but that isn’t always easy to do. [Read more…]
My second, follow-up piece was for my own blog, and included a tutorial introduction to Bayes’ theorem. Admittedly, this example is very simple and doesn’t do justice to the power of Bayesian methods, but a better example would be pretty involved, so I decided to hold off for now. (I may revisit the topic later, though, depending on time and inspiration.)
So, Bayes’ theorem reads: the probability of a hypothesis being true (based on the data and prior information) depends on the probability of the hypothesis from prior knowledge, multiplied by the likelihood of that particular data showing up, divided by the chance of the data showing up based on the priors alone. Using Bradley Efron’s example from his Science article, consider the case of a couple whose sonogram showed they were due to give birth to male twins. Given that data, they wanted to know what the chances were that the twins would be identical as opposed to fraternal — a genetic question undecidable by sonogram. [Read more…]