On Events: Rare Events

"We must rather seek for a cause, for every event whether probable or improbable must have some cause." (Polybius, "The Histories", cca. 100 BC)

"There is nothing in the nature of a miracle that should render it incredible: its credibility depends upon the nature of the evidence by which it is supported. An event of extreme probability will not necessarily command our belief unless upon a sufficiency of proof; and so an event which we may regard as highly improbable may command our belief if it is sustained by sufficient evidence. So that the credibility or incredibility of an event does not rest upon the nature of the event itself, but depends upon the nature and sufficiency of the proof which sustains it." (Charles Babbage, "Passages from the Life of a Philosopher", 1864)

"Events with a sufficiently small probability never occur, or at least we must act, in all circumstances, as if they were impossible." (Émile Borel, "Probabilities and Life", 1962)

"Most accidents in well-designed systems involve two or more events of low probability occurring in the worst possible combination." (Robert E Machol, "Principles of Operations Research", 1975)

"[…] all human beings - professional mathematicians included - are easily muddled when it comes to estimating the probabilities of rare events. Even figuring out the right question to ask can be confusing." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)

"Bell curves don't differ that much in their bells. They differ in their tails. The tails describe how frequently rare events occur. They describe whether rare events really are so rare. This leads to the saying that the devil is in the tails." (Bart Kosko, "Noise", 2006)

"A Black Swan is a highly improbable event with three principal characteristics: It is unpredictable; it carries a massive impact; and, after the fact, we concoct an explanation that makes it appear less random, and more predictable, than it was. […] The Black Swan idea is based on the structure of randomness in empirical reality. [...] the Black Swan is what we leave out of simplification." (Nassim N Taleb, “The Black Swan”, 2007)

"A forecaster should almost never ignore data, especially when she is studying rare events […]. Ignoring data is often a tip-off that the forecaster is overconfident, or is overfitting her model - that she is interested in showing off rather than trying to be accurate."  (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)

"[…] according to the bell-shaped curve the likelihood of a very-large-deviation event (a major outlier) located in the striped region appears to be very unlikely, essentially zero. The same event, though, is several thousand times more likely if it comes from a set of events obeying a fat-tailed distribution instead of the bell-shaped one." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"[…] both rarity and impact have to go into any meaningful characterization of how black any particular [black] swan happens to be." (John L Casti, "X-Events: The Collapse of Everything", 2012)

"Black Swans (capitalized) are large-scale unpredictable and irregular events of massive consequence - unpredicted by a certain observer, and such un - predictor is generally called the 'turkey' when he is both surprised and harmed by these events. [...] Black Swans hijack our brains, making us feel we 'sort of' or 'almost' predicted them, because they are retrospectively explainable. We don’t realize the role of these Swans in life because of this illusion of predictability. […] An annoying aspect of the Black Swan problem - in fact the central, and largely missed, point - is that the odds of rare events are simply not computable." (Nassim N Taleb, "Antifragile: Things that gain from disorder", 2012)

"Behavioral finance so far makes conclusions from statics not dynamics, hence misses the picture. It applies trade-offs out of context and develops the consensus that people irrationally overestimate tail risk (hence need to be 'nudged' into taking more of these exposures). But the catastrophic event is an absorbing barrier. No risky exposure can be analyzed in isolation: risks accumulate. If we ride a motorcycle, smoke, fly our own propeller plane, and join the mafia, these risks add up to a near-certain premature death. Tail risks are not a renewable resource." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"But note that any heavy tailed process, even a power law, can be described in sample (that is finite number of observations necessarily discretized) by a simple Gaussian process with changing variance, a regime switching process, or a combination of Gaussian plus a series of variable jumps (though not one where jumps are of equal size […])." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"[…] it is not merely that events in the tails of the distributions matter, happen, play a large role, etc. The point is that these events play the major role and their probabilities are not (easily) computable, not reliable for any effective use. The implication is that Black Swans do not necessarily come from fat tails; the problem can result from an incomplete assessment of tail events." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)

"Once we know something is fat-tailed, we can use heuristics to see how an exposure there reacts to random events: how much is a given unit harmed by them. It is vastly more effective to focus on being insulated from the harm of random events than try to figure them out in the required details (as we saw the inferential errors under thick tails are huge). So it is more solid, much wiser, more ethical, and more effective to focus on detection heuristics and policies rather than fabricate statistical properties." (Nassim N Taleb, "Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications" 2nd Ed., 2022)