Some take aways from Nassim Taleb “Statistical Consequences of fat tails” book. A must read for a next generation thinker.
Consequence 1
The law of large numbers, when it works, works too slowly in the real world.
Consequence 2
The mean of the distribution will rarely correspond to the sample mean; it will have a persistent small sample effect (downward or upward) particularly when the distribution is skewed (or one-tailed).
Consequence 3
Metrics such as standard deviation and variance are not useable.
Consequence 4
Beta, Sharpe Ratio and other common hackneyed financial metrics are uninformative.
Consequence 5
Robust statistics is not robust and the empirical distribution is not empirical
Consequence 6
Linear least-square regression doesn’t work (failure of the Gauss-Markov theorem).
Consequence 7
Maximum likelihood methods can work well for some parameters of the distribution
(good news).
Consequence 11
There is no such thing as a typical large deviation
Consequence 13
Large deviation theory fails to apply to thick tails. I mean, it really doesn’t apply
Practically every single economic variable and financial security is thick tailed. Of the 40,000 securities examined, not one appeared to be thin-tailed. This is the main source of failure in finance and economics.
Practically any paper in economics using covariance matrices is suspicious.
The “evidence based” approach is still too primitive to handle second order effects (and risk management) and has certainly caused way too much harm with the COVID-19 pandemic to remain useable outside of single patient issues. One of the problems is the translation between individual and collective risk (another is the mischaracterization of evidence and conflation with absence of evidence).
At the beginning of the COVID-19 pandemic, many epidemiologists innocent of probability compared the risk of death from it to that of drowning in a swimming pool. For a single individual, this might have been true (although COVID-19 turned out rapidly to be the main source of fatality in many parts, and later even caused 80% of the fatalities New York City). But conditional on having 1000 deaths, the odds of the cause being drowning in swimming pools is slim.
This is because your neighbor having COVID increases the chances that you get it, whereas your neighbor drowning in her or his swimming pool does not increase your probability of drowning (if anything, like plane crashes, it decreases other people’s chance of drowning).
This aggregation problem – joint distributions are no longer elliptical, causing the sum to be fat-tailed even when individual variables are thin-tailed. It is also discussed as a problem in ethics: by contracting the disease you cause more deaths than your own. Although the risk of death from a contagious disease can be smaller than, say, that from a car accident, it becomes psychopathic to follow “rationality” (that is, first order rationality models) as you will eventually cause systemic harm and even, eventually, certain self-harm.
Limitations of knowledge What is crucial, our limitations of knowledge apply to X not necessarily to F(X). We have no control over X, some control over F(X). In some cases a very, very large control over F(X)
The danger with the treatment of the Black Swan problem is as follows: people focus on X (“predicting X”). My point is that, although we do not understand X, we can deal with it by working on F which we can understand, while others work on predicting X which we can’t because small probabilities are incomputable, particularly in thick tailed domains. F(x) is how the end result affects you.
The probability distribution of F(X) is markedly different from that of X, particularly when F(X) is nonlinear. We need a nonlinear transformation of the distribution of X to get F(X). We had to wait until 1964 to start a discussion on “convex transformations of random variables”, Van Zwet (1964) –as the topic didn’t seem important before.
To understand more you need to listen to the vlog below. Stephen vs Nassim. Amazing. Everything will fall into place…