Нассим Талеб - The Black Swan. The Impact of the Highly Improbable

Consider that the frequency of rare events cannot be estimated from empirical observation for the very reason that they are rare . We thus need a prior model representation for that; the rarer the event, the higher the error in estimation from standard inductive methods (say, frequency sampling from counting past occurrences), hence the higher the dependence on an a priori representation that extrapolates into the space of low-probability events (which necessarily are not seen often). *

But even outside of small probabilities, the a priori problem is always present. It seems salient with respect to rare events, but it pervades probabilistic knowledge. I will present two versions I have been working on with two collaborators, Avital Pilpel, a philosopher of science (he walks fast), and Raphael Douady, a mathematician (he is sometimes a good walker, when he is not busy).

Epimenides the Cretan

Avital Pilpel and I expressed the regress argument as follows, as the epistemic problem of risk management, but the argument can be generalized to any form of probabilistic knowledge. It is a problem of self-reference by probability measures.

We can state it in the following way. If we need data to obtain a probability distribution to gauge knowledge about the future behavior of the distribution from its past results, and if, at the same time, we need a probability distribution to gauge data sufficiency and whether or not it is predictive of the future, then we face a severe regress loop. This is a problem of self-reference akin to that of Epimenides the Cretan stating whether or not Cretans are liars. Indeed, it is too uncomfortably close to the Epimenides situation, since a probability distribution is used to assess the degree of truth but cannot reflect on its own degree of truth and validity. And, unlike many problems of self-reference, those related to risk assessment have severe consequences. The problem is more acute with small probabilities.

An Undecidability Theorem

This problem of self-reference, published with Pilpel after The Black Swan , went unnoticed as such. So Raphael Douady and I re-expressed the philosophical problem mathematically, and it appears vastly more devastating in its practical implications than the Gödel problem.

Raphael is, among the people I know, perhaps the man with the greatest mathematical erudition—he may have more mathematical culture than anyone in modern times, except perhaps for his late father, Adrien Douady.

At the time of writing, we may have produced a formal proof using mathematics, and a branch of mathematics called “measure theory” that was used by the French to put rigor behind the mathematics of probability. The paper is provisionally called “Undecidability: On the inconsistency of estimating probabilities from a sample without binding a priori assumptions on the class of acceptable probabilities.”

It’s the Consequences …

Further, we in real life do not care about simple, raw probability (whether an event happens or does not happen); we worry about consequences (the size of the event; how much total destruction of lives or wealth, or other losses, will come from it; how much benefit a beneficial event will bring). Given that the less frequent the event, the more severe the consequences (just consider that the hundred-year flood is more severe, and less frequent, than the ten-year flood; the bestseller of the decade ships more copies than the bestseller of the year), our estimation of the contribution of the rare event is going to be massively faulty (contribution is probability times effect; multiply that by estimation error); and nothing can remedy it. *

So the rarer the event, the less we know about its role—and the more we need to compensate for that deficiency with an extrapolative, generalizing theory. It will lack in rigor in proportion to claims about the rarity of the event. Hence theoretical and model error are more consequential in the tails; and, the good news, some representations are more fragile than others .

I showed that this error is more severe in Extremistan, where rare events are more consequential, because of a lack of scale, or a lack of asymptotic ceiling for the random variable. In Mediocristan, by comparison, the collective effect of regular events dominates and the exceptions are rather inconsequential—we know their effect, and it is very mild because one can diversify thanks to the “law of large numbers.” Let me provide once again an illustration of Extremistan. Less than 0.25 percent of all the companies listed in the world represent around half the market capitalization, a less than minuscule percentage of novels on the planet accounts for approximately half of fiction sales, less than 0.1 percent of drugs generate a little more than half the pharmaceutical industry’s sales—and less than 0.1 percent of risky events will cause at least half the damages and losses.

From Reality to Representation *

Let me take another angle. The passage from theory to the real world presents two distinct difficulties: inverse problems and pre-asymptotics.

Inverse Problems . Recall how much more difficult it is to re-create an ice cube from the results of the puddle (reverse engineering) than to forecast the shape of the puddle. In fact, the solution is not unique: the ice cube can be of very many shapes. I have discovered that the Soviet-Harvard method of viewing the world (as opposed to the Fat Tony style) makes us commit the error of confusing the two arrows (from ice cube to puddle; from puddle to ice cube). It is another manifestation of the error of Platonicity, of thinking that the Platonic form you have in your mind is the one you are observing outside the window. We see a lot of evidence of confusion of the two arrows in the history of medicine, the rationalistic medicine based on Aristotelian teleology, which I discussed earlier. This confusion is based on the following rationale. We assume that we know the logic behind an organ, what it was made to do, and thus that we can use this logic in our treatment of the patient. It has been very hard in medicine to shed our theories of the human body. Likewise, it is easy to construct a theory in your mind, or pick it up from Harvard, then go project it on the world. Then things are very simple.

This problem of confusion of the two arrows is very severe with probability, particularly with small probabilities. *

As we showed with the undecidability theorem and the self-reference argument, in real life we do not observe probability distributions. We just observe events. So I can rephrase the results as follows: we do not know the statistical properties—until, of course, after the fact. Given a set of observations, plenty of statistical distributions can correspond to the exact same realizations—each would extrapolate differently outside the set of events from which it was derived. The inverse problem is more acute when more theories, more distributions can fit a set of data, particularly in the presence of nonlinearities or nonparsimonious distributions. †Under nonlinearities, the families of possible models/parametrization explode in numbers. ‡

But the problem gets more interesting in some domains. Recall the Casanova problem in Chapter 8. For environments that tend to produce negative Black Swans, but no positive Black Swans (these environments are called negatively skewed), the problem of small probabilities is worse. Why? Clearly, catastrophic events will be necessarily absent from the data, since the survivorship of the variable itself will depend on such effect. Thus such distributions will let the observer become prone to overestimation of stability and underestimation of potential volatility and risk.