The probability of rare or single events.

Imagine that somebody predicts that "by 2050 we will receive intelligent signals from outside the solar
system" and asks you to bet on it. If you are willing to take such a long bet, and apparently several
smart people do, it means that you are able or at least willing to assign some sort of probability to it.
But what does probability mean in such a case, dealing with a unique event/non-event far into the future?
A Bayesian statistician has some sort of answer at hand, invoking the concept of uncertainty and the
use of priors. But what does this really mean in such a situation?  And what is a frequentist supposed to
do other than to shake his head?

I assume that a rational definition of probability is actually possible in such a case and it may be possible
to reconcile both the Bayesian and frequentist philosophies: Assume that an algorithm exists which
generates different scenarios, i.e. descriptions of possible future outcomes. In simple settings, classical
physics may be sufficient and the scenarios can be generated e.g. for different initial conditions.
Indeed, engineers use such scenarios as 'failure trees', when they try to estimate the risks of a machine.
In general, the generation of such scenarios will be more involved, however, and may include past experience,
handwaving, common sense etc.
The definition of probability would then be given by counting the various scenarios and their outcomes in
frequentist fashion. The equivalence of the Bayesian prior would be the selection of an algorithm and its
parameters, used to generate the scenarios. Obviously, the results would be more reliable if the algorithm has
worked well in the past, i.e. predicted correctly. This is the point where survival of the fittest comes in.
Again, it would come down to simple counting of past success vs. failure, in the good, old frequentist way.

Unfortunately, in cases such as the intelligent signal from aliens we do not have reliable algorithms and we are
reduced to imagine a handful of scenarios. The Drake equation may be helpful, but we have no past experience
to guide us. But at least we have established what it could mean to assign a probability to such a long bet.

update: Cosma sent me this link to a blog about Estimating the probability of events that have never occurred.
It seems that the folks behind this group blog know a lot more about the topic than I do.

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