Estimation implies error, and estimates of that error can themselves be uncertain. Allowing a Gaussian model’s parameters to vary can produce a heavier-tailed distribution than a fixed-parameter model. Under particular assumptions, successive layers of uncertainty can produce power-law tails or infinite variance. Start with a Gaussian whose standard deviation σ has a proportional error a(1); the uncertainty in a(1) has an error rate a(2), and so on. The result depends on how the higher-order error rate a(n) relates to a(n−1). Constant proportional errors can produce very thick tails in the construction described; declining errors require attention to their rate of decline. The effect depends on the model and uncertainty assumptions, rather than making every small probability rise in every model.

The Fukushima example is presented as a shift from a once-per-million-years estimate to one per 30 years after allowing for layers of uncertainty. The calculation and assumptions behind that figure are not supplied here.

1 The difference between the two sides of Jensen’s inequality corresponds to a notion in information theory, the Bregman divergence. Briys, Magdalou, and Nock, 2012.

2 This further shows the defects of the notion of “Knightian uncertainty,” since all tails are uncertain under the slightest perturbation and their effect is severe in fat-tailed domains, that is, economic life.

ADDITIONAL NOTES, AFTERTHOUGHTS, AND FURTHER READING

These are both additional readings and ideas that came to me after the composition of the book, like whether God is considered robust or antifragile by theologians or the history of measurement as a sucker problem in the probability domain. As to further reading, I am avoiding the duplication of those mentioned in earlier books, particularly those concerning the philosophical problem of induction, Black Swan problems, and the psychology of uncertainty. I managed to bury some mathematical material in the text without Alexis K., the math-phobic London editor, catching me (particularly my definition of fragility in the notes for Book V and my summary derivation of “small is beautiful”). Note that there are more involved technical discussions on the Web.

Seclusion: Since The Black Swan, I’ve spent 1,150 days in physical seclusion, a soothing state of more than three hundred days a year with minimal contact with the outside world—plus twenty years of thinking about the problem of nonlinearities and nonlinear exposures. So I’ve sort of lost patience with institutional and cosmetic knowledge. Science and knowledge are convincing and deepened rigorous argument taken to its conclusion, not naive (via positiva) empiricism or fluff, which is why I refuse the commoditized (and highly gamed) journalistic idea of “reference”—rather, “further reading.” My results should not depend, and do not depend on a single paper or result, except for via negativa debunking—these are illustrative.

Charlatans: In the “fourth quadrant” paper published in International Journal of Forecasting (one of the backup documents for The Black Swan that had been sitting on the Web) I showed empirically using all economic data available that fat tails are both severe and intractable—hence all methods with “squares” don’t work with socioeconomic variables: regression, standard deviation, correlation, etc. (technically 80% of the Kurtosis in 10,000 pieces of data can come from one single observation, meaning all measures of fat tails are just sampling errors). This is a very strong via negativa statement: it means we can’t use covariance matrices—they are unreliable and uninformative. Actually just accepting fat tails would have led us to such result—no need for empiricism; I processed the data nevertheless. Now any honest scientific profession would say: “what do we do with such evidence?”—the economics and finance establishment just ignored it. A bunch of charlatans, by any scientific norm and ethical metric. Many “Nobels” (Engle, Merton, Scholes, Markowitz, Miller, Samuelson, Sharpe, and a few more) have their results grounded in such central assumptions, and all their works would evaporate otherwise. Charlatans (and fragilistas) do well in institutions. It is a matter of ethics; see notes on Book VII.

For our purpose here, I ignore any economic paper that uses regression in fat-tailed domains—as just hot air—except in some cases, such as Pritchet (2001), where the result is not impacted by fat tails.