The Daily Show on probability theory

I liked this bit on the Daily Show about the Large Hadron Collider for a bunch of reasons, mostly because John Oliver is always great.  Among other things, though, it contains a great illustration of how tricky it is, when using a Bayesian approach to probability, to choose the right prior.  That bit starts at about 3:07 and is hilariously reprised at the very end, but you should really watch the whole thing if you haven’t seen it.

Since explanations of jokes are never tedious, there’s a bit of exegesis after the jump.

The nutty guy on the Daily Show says that, if there are two possibilities (the LHC will destroy the world or it won’t), and you don’t know which one is right, then the probability of either one occuring is 50%.  John Oliver treats this idea with all the mockery and scorn it deserves, but there’s actually an interesting question in there: how should we assess the probability of a one-off future event?

In the traditional (old-fashioned) frequentist view of probability, people think of probabilities as being defined with respect to an “ensemble” of real or imagined repetitions of an experiment.  In the case of the LHC destroying the world, presumably we’d have to be talking about imagined repetitions.  Personally, I think the whole frequentist point of view is absurd, and this case is a good example of why.  It makes perfect sense to talk about the probability that the LHC will destroy the world, and it seems self-evident that when people talk about this they don’t need the entire mental apparatus of an ensemble of many imaginary copies of the Earth with different laws of physics, some of which are filled with imaginary people doomed to imaginary particle-physics-induced deaths.

In the other main approach to probabilities, known as the Bayesian approach, we admit up-front what seems obvious to me: probabilities are just statements about the incomplete knowledge of the speaker, not statements about ensembles of repeated experiments.  In the jargon of Bayesian statistics, the statement that there’s a 50-50 chance the LHC will destroy the workd is a statement about “prior probabilities” — that is,  probabilities before the experiment has been done.  If you do Bayesian statistics professionally, you have to settle in for the occasional tedious argument about how you should choose your prior probabilities.  One reasonable principle is that in the absence of information you should choose the “least informative prior,” and that seems to be what the guy on the Daily Show is arguing for.  In particular, one way to define “least informative” is as equivalent to “maximum entropy”, and 50-50 is the maximum-entropy way to assign probabilities to two possible events.

The guy is wrong to do this, of course, and it’s mildly interesting to think a bit about why.  One reason is that in assessing the prior probability you are allowed to take into account other stuff you already know.  John Ellis in that segment mentions one bit of prior knowledge, namely that cosmic rays constantly hit our atmosphere at high energies similar to what the LHC will do and haven’t yet destroyed the Earth.  When you take that prior knowledge, plus a bunch more, into account, you realize that John Ellis is right: the probability that the LHC will destroy the world is essentially zero.