4 October 2007
On Tuesday I went to a talk by Terrence Fine from Cornell University. It was one of those talks that's worth going to, if nothing else because it makes you re-visit and re-question the sort of basic assumptions that are so easy to not even notice that you're making. In this case, that basic assumption was that the mathematics of probability theory, which views probability as a real number between 0 and 1, is equally applicable to any domain where we want to reason about statistics.
Is this a sensible assumption?
As I understand it, Fine made the point that in many applied fields, what you do is start from the phenomenon to be modeled and then use the mathematical/modeling framework that is appropriate to it. In other words, you go from the applied "meaning" to the framework: e.g., if you're modeling dynamical systems, then you decide to use differential equations. What's odd in applications of probability theory, he said, is that you basically go from the mathematical theory to the meaning: we interpret the same underlying math as having different potential meanings, depending on the application and the domain.
He discussed four different applications, which are typically interpreted in different ways: physically-determined probability (e.g., statistical mechanics or quantum mechanics); frequentist probability (i.e., more data driven); subjective probability (in which probability is interpreted as degree of belief); and epistemic/logical (in which probability is used to characterize inductive reasoning in a formal language). Though I broadly agree with these distinctions, I confess I'm not getting the exact subtleties he must be making: for instance, it seems to me the interpretation of probability in statistical mechanics is arguably very different from in quantum mechanics and they should therefore not be lumped together: in statistical mechanics, the statistics of flow arise some underlying variables (i.e., the movements of individual particles), and in quantum mechanics, as I understand it, there aren't any "hidden variables" determining the probabilities as all.
But that technicality aside, the main point he made is that depending on the interpretation of probability and the application we are using it for, our standard mathematical framework -- in which we reason about probabilities using real numbers between 0 and 1 -- may be inappropriate because it is either more or less expressive than necessary. For instance, in the domain of (say) IQ, numerical probability is probably too expressive -- it is not sensible or meaningful to divide IQs by each other; all we really want is an ordering (and maybe even a partial ordering, if, as seems likely, the precision of an IQ test is low enough that small distinctions aren't meaningful). So a mathematics of probability which views it in that way, Fine argues, would be more appropriate than the standard "numerical" view.
Another example would be in quantum mechanics, where we actually observe a violation of some axioms of probability. For instance, the distributivity of union and intersection fails: P(A or B) != P(A)+P(B)-P(A and B). This is an obvious place where one would want to use a different mathematical framework, but since (as far as I know) people in quantum mechanics actually do use such a framework, I'm not sure what his point was. Other than it's a good example of the overall moral, I guess?
Anyway, the talk was interesting and thought-provoking, and I think it's a good idea to keep this point in the back of one's mind. That said, although I can see why he's arguing that different underlying mathematics might be more appropriate in some cases, I'm not convinced yet that we can conclude that using a different underlying mathematics (in the case of IQ, say) would therefore lead to new insight or help us avoid misconceptions. One of the reasons numerical probability is used so widely -- in addition to whatever historical entrenchment there is -- is that it is an indispensible tool for doing inference, reasoning about distributions, etc. It seems like replacing it with a different sort of underlying math might result in losing some of these tools (or, at the very least, require us to spend decades re-inventing new ones).
Of course, other mathematical approaches might be worth it, but at this point I don't know how well-worked out they are, and -- speaking as someone interested in the applications -- I don't know if they'd be worth the work in order to see. (They might be; I just don't know... and, of course, a pure mathematician wouldn't care about this concern, which is all to the good). Fine gave a quick sketch of some of these alternative approaches, and I got the sense that he was working on developing them but they weren't that well developed yet -- but I could be totally wrong. If anyone knows any better, or knows of good references on this sort of thing, please let us know in comments. I couldn't find anything obvious on his web page.
 I really really do not want to get into a debate about whether and to what extent IQ in general is meaningful - that question is really tangential to the point of this post, and I use IQ as illustration only. (I use it rather than something perhaps less inflammatory because it's the example Fine used).