The question of disciplinary boundaries seems to be coming up a lot lately, and Brian’s post on Gott’s Copernican principle provides yet another instance. Gott, an astrophysicist, is interested in the question of whether you can infer the future duration of a process from its present age, and this issue seems to received some discussion in philosophy journals.
It may be beneath the notice of these lofty souls, but statisticians and social scientists have actually spent a fair bit of time worrying about this question of survival analysis (also called duration analysis). For example, my labour economist colleagues at ANU were very interested in the question of how to infer the length of unemployment spells, based on observations of how long currently unemployed people had actually been unemployed. The same question arises in all sorts of contexts (crime and recidivism, working life of equipment, individual life expectancy and so on). Often, the data available is a set of incomplete durations, and you need to work out the implied survival pattern.
Given a suitably large sample (for example, the set of observations of Broadway plays, claimed as a successful application of Gott’s principle) this is a tricky technical problem, and requires some assumptions about entry rates, but raises no fundamental logical difficulties. The problem is to find a distribution that fits the data reasonably well and estimate its parameters. I don’t imagine anyone doing serious work in this field would be much impressed by Gott’s apparent belief that imposing a uniform distribution for each observation is a good way to go.
Of course, social scientists tend not to like working with a sample size of one, so the Copernicans have a bit more room to move in unique cases. Still, if you are willing to assume a functional form for your probability distribution, and there’s only one free parameter, you can calculate a maximum likelihood estimate from one data point. The arbitrary choice you make determines the confidence interval.
In Bayesian terms, picking the ML estimator is (broadly speaking) the equivalent of assuming a diffuse prior. The big problem in the Copernican approach is this assumption, which is, in effect, that you have no relevant information at all, except for your single sample observation. If the problem is of any interest at all, this assumption is almost certain to be wrong. Take the example of the likely duration of the space program. We can, at the very least, observe that NASA and its competitors have missions scheduled for years ahead, which makes very short durations much more unlikely than those derived from a uniform distribution (Brian’s examples also made this point).
The real lesson from Bayesian inference is that, with little or no sample data, even limited prior information will have a big influence on the posterior distribution. That is, if you are dealing with the kinds of cases Gott is talking about, you’re better off thinking about the problem than relying on an almost valueless statistical inference.