Jul 15, 2011

Rick’s Ramblings: Dehydrated Elephants

Well on my way to birthday “LX” (in an “XL” body), I am grateful that I had the opportunity to meet some of the greatest probabilists of our time. Monroe Donsker, with his wild and crazy eyebrows, took a hiatus from academia between inventing his invariance principle and working with Varadhan on large deviations in order to serve in the diplomatic corps. One night at a party in Spain, while hearing about the exodus of talented individuals from that country he quipped, “Oh you mean, the drain in Spain is mainly on the brain.” This has to be one of the greatest one-liners of all time, even if he was later reprimanded for it.
My second dear departed hero, who gives the title to this piece, is Marc Kac, a self-described “simple pole.” Throughout his life, he railed against dehydrated elephants. Naively, I thought that in the age of the Internet I could find a copy of the classic cartoon with the punch-line: “I’m not sure what you can do with a dehydrated elephant, but it is nice to see that it can be done.” In my opinion, this phrase applies to many of the articles that the IMS publishes its two probability journals. The problem with our publications is best illustrated by a classic Mark Kac line that complains about people who start with the answers and then figure out what is the question. The joke begins, “The answer is: 9w.” After a sufficiently pregnant pause, the question is revealed: “Do you spell your name with a V, Mr. Wagner?”(“Nein, W”). If you want to learn more about the man behind these jokes, check out Kac’s autobiography, Enigmas of Chance.
Returning to my diatribe, one of the problems about what is published in the Annals of Applied Probability lies in the dictum applied by referees: “if it’s not hard, it’s not good.” It seems to me that one should give at least equal weight to the question: does the paper say something interesting about the application? To quote my academic godfather, Kai Lai Chung, from the preface of his book on Markov Chains: “Mathematicians are more inclined to build fire stations than to put out fires.” Given the content of our journals, the quote should be updated to: “Once we have a blueprint for one fire station, there is no need to actually build it or to engage in the boring enterprise of putting out fires.”
This is not to say that theoretical probability is dead and buried. The preview of Alice Guionnet’s Medallion lecture on random matrices [in the previous issue] is very exciting. At my new home in Durham, we recently had a 2011 revival of the Southeastern Probability Conference. The first conference brought an excellent group of speakers: Timo Seppäläinen (Wisconsin), Jeremy Quastel (Toronto), Lionel Levine (MIT), Pete Kramer (RPI) , Davar Khoshnevisan (Utah), Sourav Chatterjee (NYU), and Gérard Ben Arous (NYU). As I sat there and watched the talks, I could only say WOW. Of course, much of the work presented at the SEPC had a motivation coming from physics or biology, but some talks like Lionel Levine’s lecture were mathematically beautiful solutions.
Turning—as I inevitably do in this column—to discussing my own research, one of the great things about coming to Duke a year ago has been the fact that the medical school is on campus, rather than 4+ hours away from “centrally isolated” Ithaca (another Marc Kac phrase that could easily be applied to SAMSI’s location in the Research Triangle Park).
One of my new collaborations with local medical researchers involves a study of ovarian cancer. In 2010, an estimated 21,800 new cases developed in the US with 13,850 deaths. As the numbers might suggest this is a deadly disease with a rapid time course. The five-year survival rate is 30%, due primarily to the fact that the tumors which start in the ovary and fallopian tubes do not have significant physical barriers to metastasis into the peritoneal cavity.
With a very talented Duke undergraduate, Kaveh Danesh, I have been working to apply a mechanistic model as an alternative to the phenomenological models that have women hopping from clinical stage I to II to III to IV according to a Markov chain. Our models are multitype branching processes in which type i individuals give birth at rate ai, die at rate bi, and mutate to type i+1 at rate ui+1. Before you yawn and say that studying these models is not hard enough to be worth it, put down this issue of the Bulletin and try to figure out what the asymptotic behavior of Z1 has to do with one-sided stable laws, an observation that provides useful insights into tumor heterogeneity and yields remarkably simple explicit formulas for various quantities.
Returning to the theme of dehydrated elephants, in this project, Kaveh and I have benefited from email exchanges with Vladis Pipiras, Sid Resnick, and Gennady Samorodnitsky, hydrated humans who helped us navigate the voluminous literature of stable laws. Let Λ be a one-sided stable law with index α. Why the heck Peter Brockwell and Bruce Brown in an article in the ZfW in 1978 decided to look at Λ–α and to produce a power series for its density that has two gamma functions in the denominator, I don’t know. Probably, they just wrote down the formula because they had the tools to do so. However, it has been useful for making connections with Kaplan-Meier estimates of the size of the primary tumor when the patient enters the deadly stage III of the disease.
To wrap up this Ramble, the take-home message, which I learned almost twenty-five years ago in work with Simon Levin is this: when your problems come from real questions, then the answers lead you to a territory that is more interesting than when you start with the answer and figure out the question.
Grinding my other ax: probability theory was designed to solve problems, but much of what is published is like poetry in Esperanto, an esoteric art form that can be appreciated only by a handful of people. Is this really what we want in our “yellow journalism”?

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