Nassim Taleb has published a draft version of his Incerto [STATISTICAL CONSEQUENCES OF FAT TAILS [TECHNICAL INCERTO COLLECTION] which includes a chapter with a non-technical look at his work on what living in a world where many items / distributions are “fat or long tail”. (Link end via Academia.edu). It also has some interesting book design for a text book type work.
One observation is that many many distributions are fat tailed.
Eg Earthquakes, Wealth, many human constructs like stock markets.
He then shows that standard statistics fail under most of these other distributions and even some of these where the central limit theorem may work the number of observations needed for it to work are beyond what humans have.
Eg if you have 148 months of market data known to be fat tailed, none of the standard statistics tell you anything useful about the future.
Or even with a few hundred years of earthquake data, we don’t know when (to say the the nearest year or even decade) or how large the next San Francisco earth quake will be.
Another observation is that events in fat tails have, or can often have, large consequences.
For instance -
A stock market crash of -50 percent of more, a large earthquake
And so, putting together those observations:
Unpredictable large consequence events that standard statistics fail to model
The suggested solutions are to be anti-fragile or to cope well with the large consequences rather than the fruitless task of preventing or predicting them exactly. (One can predict an event will happen eg earthquake but typically not exactly when; although maybe there can other pattern warning signs one can look for on a short run basis)
I would add his and other observations about complex systems - that changes (and some times seemingly small and inconsequential changes) can have large (and negative / unexpected) consequences (this is via the thinking that Mandlebrot articulated - “butterfly flaps its wings…”)
This thinking leads to a few intriguing solutions and ideas
On climate, we should take action because climate is a complex system and too much human intervention (pollution) can tip it over to a large negative consequence (no planet B)
A diversity of small local systems are anti-fragile versus one large system (one big bank systemically a bad idea vs many small banks; diversity of crop varieties is better than only one or two kinds (even and perhaps with more reason, if they’ve been engineered to be resistant in any one aspect); variety of assets, skills and incomes streams are better than a single one)
On understanding different types of distribution:
“…Let us randomly select two people in Mediocristan; assume we obtain a (very un- likely) combined height of 4.1 meters – a tail event. According to the Gaussian dis- tribution (or, rather its one-tailed siblings), the most likely combination of the two heights is 2.05 meters and 2.05 meters. Not 10 centimeters and 4 meters.”
“Let us now move to Extremistan and randomly select two people with combined wealth of $ 36 million. The most likely combination is not $18 million and $ 18 million. It should be approximately $ 35,999,000 and $ 1,000.
This highlights the crisp distinction between the two domains; for the class of subex- ponential distributions, ruin is more likely to come from a single extreme event than from a series of bad episodes. This logic underpins classical risk theory as outlined by the actuary Filip Lundberg early in the 20th Century [123] and formalized in the 1930s by Harald Cramer[44], but forgotten by economists in recent times. For insura- bility, losses need to be more likely to come from many events than a single one, thus allowing for diversification,
This indicates that insurance can only work in Mediocristan; you should never write an uncapped insurance contract if there is a risk of catastrophe. The point is called the catastrophe principle.
As we saw earlier, with fat tailed distributions, extreme events away from the centre of the distribution play a very large role. Black Swans are not "more frequent" (as it is commonly misinterpreted), they are more consequential. The fattest tail distribution has just one very large extreme deviation, rather than many departures form the norm. “
Many of themes are already explored in his on going series of work but this draft work adds further examples and some maths backing it up.
Link here to it on Academia.Edu