Nassim Taleb: How I Spent My 2009 -- Thinking, Drinking Wine, Upstaging Kurt Godel, etc.
My friend Nassim Taleb has out a typically discursive, smart and did-he-really-say-that musing up as a year-end comment on his non-blog blog. He talks about the merits of a 4-hour-workweek, his increasing productivity now that he makes fun of politicians and spends less time on the Interwebs, etc. -- and then there is this about upstaging that Kurt Godel guy:Raphael Douady and I re-expressed the philosophical problem mathematically, and it appears vastly more devastating in its implication than the Gödel problem. At the time of writing, we produced a formal proof using mathematics, and a branch of mathematics called "measure theory" that was used by the French to put rigor behind mathematics of probability. The paper is temporarily called Undecidability theorem of probabilistic measures: On the inconsistency of estimating probabilites from a sample without binding a priori assumptions on the class of acceptable probabilities.
Take the basic question of probabilistic inference as follows: "given a set of observations, what are the most probable probability measures it was drawn from?" or else "what is the conditional probability that a given probability measure was the one drawn from, knowing the sample?" This question requires that an a priori probability measure be given on the space Ã(Â) of probability measures on Â. (a "probability measure" is a positive measure with total weight summing up to 1). In order to weaken this assumption, one can simply ask what is the "probability density function" (PDF) on the space Ã(Â)implicitly implied by a sample. Even this question requires a "natural" measure, such as the "Lebesgue measure" on Ã(Â), even though such a measure would not be finite (the total weight is infinite, but it is possible to compute a Radon-Nikodym density with respect to this measure).The problem is that there is no "natural" probability measure on the space Ã(Â) because, this space being infinite dimensional; thanks to Riesz theorem, it is not locally compact, and this fact prevents the existence of a Lebesgue measure on the space itself. Consequently, unless one make a extremely strong a priori assumption that the set of "acceptable" measures is fully described by a finite number of parameters, the "basic" statistical question is meaningless.