Proof in the Flesh
Taleb argues that rare-event probabilities estimated from limited economic data can be highly unstable. His concern is that sampling error and uncertain tail parameters can make apparently precise risk estimates unreliable.
THE IDEA
Proof in the Flesh
There is no reliable way to compute small probabilities. I argued philosophically the difficulty of computing the odds of rare events. Using almost all available economic data—and I used economic data because that’s where the clean data was—I showed the impossibility of computing from the data the measure of how far away from the Gaussian one was. There is a measure called kurtosis that the reader does not need to bother with, but that represents “how fat the tails are,” that is, how much rare events play a role. Well, often, with ten thousand pieces of data, forty years of daily observations, one single observation represents 90 percent of the kurtosis! Sampling error is too large for any statistical inference about how non-Gaussian something is, meaning that if you miss a single number, you miss the whole thing. The instability of the kurtosis implies that a certain class of statistical measures should be totally disallowed. This proves that everything relying on “standard deviation,” “variance,” “least square deviation,” etc., is bogus.
Further, I also showed that it is impossible to use fractals to get acceptably precise probabilities—simply because a very small change in what I called the “tail exponent” in Chapter 16, coming from observation error, would make the probabilities change by a factor of 10, perhaps more.
Implication: the need to avoid exposure to small probabilities in a certain domain. We simply cannot compute them.
When to use it
Risk management: test the sensitivity of rare-event estimates before treating a precise output as dependable.
Investment strategy: Prefer robust portfolios that minimize exposure to tail risks rather than optimizing based on expected returns.
Policy analysis: examine exposure to catastrophic outcomes alongside uncertain probabilities, intervention costs, and alternative safeguards.
FROM THEORY TO EVERYDAY LIFE
In practice
Economic data and kurtosis
Taleb reports that, in some economic data, one observation among 10,000 daily observations over 40 years can account for 90% of the kurtosis. The example illustrates sensitivity to rare observations, rather than proving that every small probability is unknowable.
Insurance and tail exponents
When estimating a 100-year flood probability with a heavy-tailed model, test how sensitive the estimate is to the fitted tail exponent. Large sensitivity limits the precision justified by the data.
A MOMENT TO REFLECT
Put it to work.
How does the instability of kurtosis affect your trust in statistical models that rely on variance?
Can you identify a real-world situation where a single rare event dominates the statistics, and what does that imply for decision-making?
How should finance, insurance, or public-safety decisions change when a rare-event probability is highly uncertain?
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)