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July 18, 2007

A very strong prior

In A Survival Imperative for Space Colonization, John Tierney discusses the ultra-simple predictions of Dr. Gott of Princeton's Physics program. The question of interest is how long with something last. He then makes an assumption which seems weak at first but turns out to be humongous. He assumes "there is nothing special about the particular moment that you're observing this phenomenon".

So if something has existed for X units of time then the probability it will die (end, finish, whatever) in the next unit of time is 1 / (X+1) . Through an argument of mathematical symmetry, this suggests a 95 per cent likelihood, the future of a thing will be between 1/39 and 39 times as long as its past (2.5% on each side).

First, this is a neat parlor trick, because like a good con, once you agree to the premise you are in for the ride until the end. The problem is you are taking a datum, and enforcing a very power prior distribution on the parameter of interest. Saying that their is nothing special about today's observation is rarely true. For most events we'd like to make predictions about, there are other similar events, qualitative and quantitative data observed over time and distance, and theoretical and empirical reasons that they take a particular form. To ignore that is to discard information that not would allow you to make a more accurate prediction but also invalidates the nothing special assumption.

Second, you generate huge intervals around your life expectancy estimates, so one would be rather surprised if they happened to be wrong.

Third, is this technique falsifiable or disreputable? How many bad predictions would this have to make to teach us not to do it, as long as one can point out situations when it has worked?

Be careful making predictions when the only information you is a piddling single observation. Sure you can use a strong prior assumption about the distribution of the life expectancy for the thing you've been observing. However, since you have only one observation, all you are doing is blending your preconceptions with a single piece of evidence, and that's not going to be much different than going with your assumption in the first place.

Addenda:
Brian over at crooked timber has broke down the algebra to solve for arbitrary interval probabilities.

Posted by OneEyedMan at July 18, 2007 12:15 AM

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