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
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
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,
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
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.
would come down to simple counting of past success vs. failure, in the
good, old frequentist
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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