The Black Swan reloaded

In The Black Swan Nassim N. Thaleb suggests extreme uncertainty, the Black Swans, explain almost everything about our world. Unfortunately the book is vague and full of endless passages going nowhere. Here I give you a concise version from my point of view.

Exponential distributions: Thaleb’s white swans

You want to know how fast COVID-19 passes along a chain of n+1 individuals. You learn about the generation time interval, the time it takes from the infection of one individual to the next. You’re told the generation time interval is a random variable with a gamma distribution with shape parameter a and scale parameter T, which is quite realistic. The time it takes COVID-19 to pass along the chain of n+1 individuals is the sum of the n generation time intervals associated with each of the n transmission events: Sn=T1+T2+...+Tn. Now you remember the central limit theorem (Feller 1971, ): when n is large the distribution of Sn can be approximated by a Normal distribution with mean(Sn)= and variance(Sn)=2, where μ and σ2 are the mean and variance of your variable. For the generation time intervals μ=aT and σ2=aT2 and therefore you obtain mean(Sn)=naT and variance(Sn)=naT2. Next you want to calculate the pX, the time interval that will not be exceeded in X% of the transmission chains containing n+1 individuals. The pX is an informative variable if you’re in the risk business. In this example it tells you that, with a X% confidence, any individual infected after the pX belongs to a transmission chain longer than n+1. Since Sn is approximated by a normal distribution then pXnormal=cn(X)T, where cn(X) is the solution to the equation X=100×{1+erf[(cn-na))/(2na)½]}/2 and erf(x) is the error function.

Let’s check if your calculations are correct. The sum of random variables with a gamma distribution is a gamma distribution with the same scale parameter and the shape parameter multiplied by the number of variables. Therefore the distribution of Sn is a gamma distribution, with shape parameter na, scale parameter T, mean(Sn)=naT and variance(Sn)=naT2. That’s exactly the mean and variance you obtained using the central limit theorem. What about the pX? From the gamma distribution we obtain pXgamma = gn(X)T, where gn(X) is the solution of the equation X×Γ(na)=100×γ(na,gn(X)), Γ(x) is the gamma function and γ(x,y) is the incomplete gamma function. Let’s take the ratio to compare the results pXnormal/ pXgamma = cn(X)/gn(X). I have calculated this ratio for you. You can also do it yourself with the Python script at the bottom. If you want 80% confidence, the ratio is close to 1.00 with an error below 5%. The central limit theorem gives you a very good estimate of the p80. If you want a higher 95% confidence, the ratio is again close to one but now the error can get as high as 9% for small n×a. The more confidence you want, the further you get into the distribution tail and the lower the accuracy of the central limit approximation. 

No matter how naive this example is, it is real. Those calculations are informative to understand the time scales associated with the spreading of COVID-19. One may argue that the distribution of generation time intervals is not a gamma distribution but some other distribution. Fine, the only thing that matters is evidence indicating that the distribution is exponentially bounded. If that is the case, we have a go. We can deploy the central limit theorem.

Bottom lines:

  • Some real problems are characterized by exponentially bounded distributions.

  • For exponentially bounded distributions you can apply the central limit theorem (check for dependency as well). 


Subexponential distributions: Thaleb’s gray swans

You’re marketing a product and you want to know how many direct friends are associated with n individuals picked at random from the population. You find out that the distribution of the number of direct friends per individual has a fat tail. Not sure about what is the exact shape, but you’re certain it is fatter than any exponential distribution. Calamity! You cannot apply the central limit theorem. What then? Fortunately for you, the “useless” theoreticians have studied these distributions. They call them subexponential distributions.

By definition, the probability of large values drawn from a subexponential distribution cannot  be bounded by any exponential distribution. If you attempt to fit the tail of the distribution by an exponential function the points will lie above the fit beyond a certain value. Some illustrious examples of subexponential distributions are the log-normal, Weibull, Pareto and Cauchy distributions. It does not make sense to approximate the sum of random variables with a subexponential distribution by the Normal distribution. To start with, your estimate of the sample variance may get bigger the more samples you collect. What then? If you take the sum of two quantities x and y extracted from the same subexponential distribution then the distribution tail of x+y is approximately equal to that of max(x,y) (Foss 2013). You would think that pulling random variables together reduces the fluctuations when you’re actually fishing for the extreme. When you draw numbers from a subexponential distribution then with high probability one of the draws will be much larger than the rest. You may rush to call it an outlier, but you should not. That property, the dominance of one of the draws, is the rule when dealing with subexponential distributions.

The lost of 3,000 people in the aftermath of the September 2011 terrorist attacks was extreme.  Yet, it is not an outlier. It is the expectation given the power law distribution of death tall from terrorist attacks (Clauset 2007, Bohorquez 2009). Terrorist attacks are the consequence of extreme wrath combined with the lack of empathy. There will always be a fraction of the population lacking empathy. We should then focus on reducing inequality and oppression, the major drivers of wrath. Unfortunately we haven’t learned to live in harmony and the conditions fueling wrath still remain.

Having extreme events is not always bad. In the marketing example above, you want to reach as many people as possible. Knowing that you're dealing with a subexponential distribution of peoples connectivity is a blessing. You go further and find out that the connectivity distribution of social networks is well fitted by the power law k-a, where k≥1 represents the connectivity per person and a>1 is the power law exponent. Because of the subexponential property Probability( k1+k2+...+kn x) ≈ Probability( max(k1,k2,...,kn) ≤ x) and for the power law Probability( max(x1,x2,...,xn) ≤ x) = (1-x1-a)n ≈ exp(-nx1-a). The latter is the Fréchet distribution, the limit distribution of the maximum of random variables with a fat tailed distribution (Coles 2001). Using that equation you can calculate the pX, X% = 100% Probability(k1+k2+...+kn≤pX), obtaining pX = (-n/log(X/100))1/(a-1). For example, for a=2 you get p80 = 4.5×n. With 80% confidence the sum of direct connections of n individuals is at most 4.5×n. That means you’ll reach 450 individuals through the direct connections of your 100 targets. In contrast, if a=3/2 (a fatter tail), you get p80 = 20×n2. You target 10 people and you could reach 2,000 other people through their direct connections. If you’re into marketing you love social networks fat tails. You have learned one more thing. A slight change in the power law exponent from 2 to 3/2 changed the forecast from 450 to 2,000. Knowing that you are in the subexponential scenario is not sufficient. You need to make an effort to determine the shape of the distribution tail. 

Bottom lines:

  • With subexponential distributions extreme is the normal.

  • Pulling together entities with subexponential distributions selects for extremes.

  • Your forecast is as good as your estimate of the distribution tail.

Phase transitions: Thaleb’s black swans

You are in the tulips business and you happen to live in Amsterdam during the tulip mania period. You may have lost all your wealth. The tulip mania is an example of financial contagion: A trading instrument that becomes popular, its use spreads like a virus and eventually crashes due to its shortcomings. Financial contagions, like the COVID-19 pandemics, belong to the most extreme scenario of phase transitions. The term phase transition is borrowed from Physics, where it was used at first to describe the transition between the phases of matter (gas, liquid, solid, plasma, ...). When you warm ice it turns into water (solid to liquid). When you boil water it turns into vapor (liquid to gas). Phase transition happens following gradual changes of a control parameter. Temperature in our previous example. Ice remains ice as long as temperature remains below 0°C. Yet, it turns into water if warmed one more degree. At the microscopic level it is the same, molecules of water. At the macroscopic level the differences are extreme: solid, liquid or vapor.

You may have heard about pyramid schemes or cut in one. Recruit people into the scheme and ask them to do the same. If you recruit R people, each of them R people and so on then the magic of geometric series 1+R+R2+R4+...+Rn = (Rn+1-1)/(R+1). The number of recruits grows exponentially (Rn) with the depth (n) of the pyramid. Well, no so fast. That is true if and only if R, what we call the reproductive number, is greater than 1. Eventually people run out of new recruits and R falls below 1, departing from the exponential growth. The founders will be at the root of a large tree. The late adopters will have none or a few leafs above them. If the pyramid was conceived for the distribution and sales of a product then the founders will make money while the leafs will make little or none. The distribution of narcotics works that way. The cartel becomes rich and the dead end distributors die or go to jail.

Another example is the spread of infectious diseases, such as the SARS-Cov-2 virus responsible for the ongoing COVID-19 epidemics. Similar to the pyramid scheme, the spread of a virus is characterized by its reproductive number: the number of other people infected by an individual already infected by the virus.  As long as the reproductive number R<1 we are told we’re safe, the virus spread is localized to small groups. In contrast, when R>1 we are in trouble, the virus is spreading widely causing a global outbreak. The phases in this example are the contained spread (R<1 or subcritical) and the pandemic scenario (R>1 or supercritical). We could think of the spreading of a financial instrument as an infectious disease among finance practitioners and quantify its reproductive number R. R<1 implies that the financial instrument is adopted by a few and does not cause any financial distress. In contrast, we should worry about R>1, because it means the financial instrument is spreading widely and its shortcomings will end causing a financial crash if not stopped in time.

When we look at the stock market we see a mix of both. There are many sub-critical financial contagions that are responsible for the constant fluctuations of stock prices. Then, from time to time, there is a dramatic drop in stock prices as a result of a supercritical financial contagion. The empiricists Nassim N. Thaleb just looks at the distribution of stock market changes. He sees a bimodal distribution of changes. A distribution for small and intermediate stock price changes and a big bump for the financial crashes. Nassim N.N. Thaleb goes ahead and calls the bump the Black Swans. So far so good, that's just wording. However, he does not stop there. He adds that we have no way to predict their occurrence. Here I disagree. Every phase transition has a control parameter. That control parameter can be monitored and be subject to mitigation strategies to avoid the phase transition to happen. That is how we work against infectious disease outbreaks and that is how we can work against financial contagions.

One could argue that the everlasting COVID-19 pandemic is a demonstration of our inability to predict phase transitions, Black Swans if you wish. I disagree. It is a demonstration of our inability to deploy our scientific knowledge. There were early warnings, we just didn’t act with the required response. At the beginning WHO and other health organizations advised against the use of face masks. The advice was based on the absence of evidence of airborne transmission. There was no evidence for absence of airborne transmission either, which should have resulted in the recommendation of face mask use. The experience with the SARS-2002 and MERS-2012 outbreaks was there. All the data about superspreaders was there. The information about worldwide spread through air travel was there. Yet, we didn’t have government protocols to use it.

There are complex systems hiding a whole array of interconnected phase transitions. Gaia, the dynamical system composed of living and nonliving things on earth, is a good example (Lovelock 1979). We are observing the melting of glaciers and icebergs as the Earth gets hotter. We are experiencing extreme weather events at a higher frequency than in the past. We are witnessing extinction of species due to our destruction of their habitats or due to excessive harvesting. Extinction itself is a phase transition, when the species reproductive number falls below 1. The gradual increase of the Earth's atmospheric temperature is pushing Gaia across these phase transitions. I fear what could come next. The solution is in front of us. Halt or even revert global warming. Here again the problem is not lack of understanding or uncertainty. The problem is lack of the required action.

Bottom lines:

  • When dealing with phase transitions there are two distributions, one for each phase.

  • The control parameter of the phase transition tells us when we are transitioning to a supercritical phase and what interventions are getting us out of there.

  • If you think you’re in this scenario, find the control parameter and keep your eye on it.


We understand uncertainty very well and we know what to do about it. Uncertainty is not the problem. We're the problem. Our lack of empathy for ourselves and Gaia is the problem. Our lack of action to protect ourselves and Gaia is the problem.

Alexei Vazquez


Aaron Clauset, Maxwell Young, Skrede Gleditsch, On the Frequency of Severe Terrorist Events. Journal of Conflict Resolution 51:58-87 (2007)

Juan C. Bohorquez, Sean Gourley, Alexander A. Dixon, Michael Spagat and Neil F. Johnson, Common ecology quantifies human insurgency. Nature 462, 911–914 (2009)

Stuart Coles, An Introduction to Statistical Modeling of Extreme Values (Springer-Verlag, London, 2001) 

Serguey Foss, Dmitry Korshunov and Stan Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions (Springer, New york, 2013).

James Lovelock, Gaia: A New Look at Life on Earth (Oxford University Press, Oxford, 1979).

Nassim N. Thaleb, The Black Swan (Random House, New York, 2010).

William Feller, An Introduction to Probability Theory and its Applications, Volume II (John Wiley & Sons, New York, 1971).

Python code for your own fun

Central limit theorem for gamma distributions

from sympy.solvers import nsolve

from sympy import Symbol, gamma, lowergamma, erf

x = Symbol('x')

for confidence in [0.8,0.95]:

    for a in [1,2,3]:

        for n in [2,3,4,5]:






Max vs sum for log-normal distributions

import numpy as np

import matplotlib.pyplot as plt


fig, ax = plt.subplots()

bins = np.logspace(1, 7, 7, base = 2)

widths = (bins[1:] - bins[:-1])


hist = np.histogram(x.sum(axis=1), bins=bins)

count = hist[0]/(n*widths)

ax.plot(bins[:-1],count, label='sum')

hist = np.histogram(x.max(axis=1), bins=bins)

count = hist[0]/(n*widths)

ax.plot(bins[:-1],count, '--', label='max')


ax.legend(loc='upper right')


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