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Let us start with the inverse problem of rare events and proceed
with a simple, nonmathematical argument. In August 2007, The Wall Street
Journal published a statement by one financial economist, expressing his
surprise that financial markets experienced a string of events that "would
happen once in 10,000 years". A portrait of the gentleman accompanying the
article revealed that he was considerably  younger than 10,000 years; it
is therefore fair to assume that he was not drawing his inference from his own
empirical experience (and not from history at large), but from some theoretical
model that produces the risk of rare events, or what he perceived to be rare
events.

Alas, the rarer
the event, the more theory you need (since we don't observe it). So the rarer the event, the worse its
inverse problem. And theories are fragile (just think of Doctor
Bernanke).

The tragedy is as follows. Suppose that you are deriving
probabilities of future occurrences from the data, assuming (generously) that
the past is representative of the future. Now, say that you estimate that an
event happens every 1,000 days. You will need a lot more data than 1,000 days
to ascertain its frequency, say 3,000 days. Now, what if the event happens once
every 5,000 days? The estimation of this probability requires some larger
number, 15,000 or more. The smaller the probability, the more observations you
need, and the greater the estimation error for a set number of observations.
Therefore, to estimate a rare event you need a sample that is larger and larger
in inverse proportion to the occurrence of the event.

If small probability events carry large
impacts, and (at the same time) these small probability events are more
difficult to compute from past data itself, then: our empirical knowledge about the potential contribution—or role—of rare events (probability × consequence) is
inversely proportional to their impact. This is why we should worry in the
fourth quadrant!

For rare
events, the confirmation bias (the tendency, Bernanke-style, of finding samples
that confirm your opinion, not those that disconfirm it) is very costly and
very distorting. Why? Most of histories of Black Swan prone events is going to
be Black Swan free! Most samples will not reveal the black swans—except
after if you are hit with them, in which case you will not be in a position to
discuss them. Indeed I show with 40 years of data that past Black Swans do not predict future Black Swans
in socio-economic life.