Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications

by Nassim Nicholas Taleb

The book investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin tailed to fat tailed distributions requires more than “changing the color of the dress.” Traditional asymptotics deal mainly with either n=1 or n=∞, and the real world is in between, under the “laws of the medium numbers”–which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few - The sample mean is rarely in line with the population mean, with effect on “naïve empiricism,” but can be sometimes be estimated via parametric methods. - The “empirical distribution” is rarely empirical. - Parameter uncertainty has compounding effects on statistical metrics. - Dimension reduction (principal components) fails. - Inequality estimators (Gini or quantile contributions) are not additive and produce wrong results. - Many “biases” found in psychology become entirely rational under more sophisticated probability distributions. - Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.

Genres: Mathematics, Finance, Economics, Philosophy, Nonfiction, Science, Business

441 pages, Hardcover

Published June 30, 2020

About the author

Nassim Nicholas Taleb spent 21 years as a risk taker (quantitative trader) before becoming a flaneur and researcher in philosophical, mathematical and (mostly) practical problems with probability. 


Taleb is the author of a multivolume essay, the Incerto (The Black Swan, Fooled by Randomness, Antifragile, and Skin in the Game) an investigation of opacity, luck, uncertainty, probability, human error, risk, and decision making when we don’t understand the world, expressed in the form of a personal essay with autobiographical sections, stories, parables, and philosophical, historical, and scientic discussions in nonover lapping volumes that can be accessed in any order.

In addition to his trader life, Taleb has also written, as a backup of the Incerto, more than 50 scholarly papers in statistical physics, statistics, philosophy, ethics, economics, international affairs, and quantitative finance, all around the notion of risk and probability.

Taleb is currently Distinguished Professor of Risk Engineering at NYU's Tandon School of Engineering (only a quarter time position). His current focus is on the properties of systems that can handle disorder ("antifragile").

Taleb believes that prizes, honorary degrees, awards, and ceremonialism debase knowledge by turning it into a spectator sport.

See Wikipedia for more details.

Reviews

[Richard Zhu · Rating 4 out of 5]

TBH I expected a lot of rambling -- I did get the rambling, a smattering of straight dunks on famed economists, but also a bunch of interesting technical commentary on extremal value theory, fat-tailed distributions, and their connections to stock market phenomena and rationality.

Examples:
- ReLU/ramp activation functions in ML are call/put options. Hence, one can approximate any nonlinear payoff function with a collection of call + put options.
- You know those people that want to have "well-calibrated" sense of probability? Yeah, that's all bullshit - most people should be sensitive to the payoff over the outcome, not the probability of the outcome. Unless you make a living off of binary predictions and your success rate % (most people don't), one should therefore be behaviorally "irrational".
- Since it's easier to reason about exposures arising from ramp/ReLu/call/put options as opposed to binary options, one sees highly-liquid markets in the former but not in the latter (binary options are mostly banned, prediction markets haven't really taken off).
- Fun observation (which made Taleb's career): the distribution of positive returns in the S&P 500 index looks different than that for negative returns. In fact, the negative ones look like a power law, but the positive ones are something else entirely. Leveraging this into a strategy: lose small amounts for a long time by being short the market with options, and then make huge gains when (insert black swan here) happens.

"We hold that calibration in frequency space is an academic exercise (in the bad sense of the word) that mistracks real life outcomes outside narrow binary bets."

Make sure to brush up on your probability or just skip over the equations and right to the images.

[Clive F · Rating 3 out of 5]

I've just started reading this, and wanted to say TYPOS before I forgot. Not that I could forget, there's one every couple of pages. Which is annoying, as there is clearly good stuff in here, from which this detracts - those formulae that I can't quite follow through, is that a typo, an error by the author, or just me being dim? Hmm... I shall report more as I get further in...

Right, we've finished it now. Taleb is certainly opinionated - very opinionated about other people and their views, at times, and personally I felt this rather detracted from the writing. Keep it for the Twitter wars, mate, really.

The book is a compilation of a number of papers, some of them quite technical, about the subject matter. Throughout, I ran into the issue mentioned in the first paragraph - typos are rife, and quite disturbing in the middle of formulae or proofs.

Much of what Taleb says, however, is important, even if he does rather aim at getting in his retribution first against his perceived opposition. He's quite right, fundamentally, about the difference between probability and reality: in the real world, there are no probabilities - things happen, or they don't. Probability expresses our uncertainty, not a real-world property of objects (perhaps quantum mechanics aside - although by some interpretations, this is just as true there, too). So we need to be very very careful when we apply simple to manipulate theorems of probability (such as the Gaussian distribution) to real-world problems, especially when we are making high-impact decisions based on the result of those manipulations. Examples abound, but the most dramatic is perhaps the Value at Risk (VaR) calculations that were supposed to keep our financial institutions "safe" during market turbulence, and which so spectacularly failed to do so in the 2008 financial crisis.

Tough for me to review this book usefully. I'm pretty mathematically smart, but I still couldn't follow all of the arguments. I'm sure others with more of the appropriate background could do so, but in this case I'm not sure it went far enough. Replacing VaR by Extreme Value Theory is all very well to talk about in principle, but you're going to need a lot more detail to make it practical, for example.

Three stars, in the end, for the sound argument, counterbalanced by TYPOS and unnecessary sarcasm from the author.

[Fernand · Rating 4 out of 5]

Read the pre-print. Blew my mind. This was for me a better explanation of NNT’s messages than his pure prose books if you have a good background on statistics and random processes.

[Georvic · Rating 5 out of 5]

It was extremely illuminating. I come from having a background in data science and I was in the middle of a project that involved portfolio optimization theory. This book taught me the limits of this theory quite well. Of course, this book is a lot more general than that, so you can expect to find it useful for whatever you are doing, as long as you want to learn about the limits of the statistics that is typically learned and used within your domain. However, I must say that I found this book extremely hard to read. I feel the author assumes a lot. However, in my case, this problem was solved just by painstakingly googling a lot.

[William Bies · Rating 3 out of 5]

While it may seem obvious to reflect, as a matter of theory, that the initial attempt at scientific comprehension in a novel domain of experience will, more often than not, have to be revised in light of later developments – perhaps drastically –, in practice, simplified toy models often endure long after they can have been seen to be inadequate. Consider, for instance, how long it took for the modern view of chaos theory to establish itself, against deeply ingrained habits of thought on the part of physicists. Just why was this? It happens that the linear approximation (in which chaotic behavior does not occur) lends itself to a completely satisfactory understanding of a wide domain of phenomena. For example, the normal modes of vibration of a solid body or the principal axis theorem concerning how it tumbles, if tossed into the air (it can rotate stably about the axes either of least or of greatest moment of inertia, but rotation about the intermediate axis is unstable – this fact can readily be observed by throwing a book up and down). Hence, its explanatory successes lulled everyone into the complacent anticipation that it could continue to serve as paradigmatic, even as the progress of science brought new phenomena into prominence, which it is ill-equipped to handle. Thus, when a cadre of intrepid investigators who know better manage to overthrow the old regime, it will come as little surprise that they share a heightened sense of the import of the revolution they bring about.

Now, the world of finance has witnessed a revolution similar to what happened in physics with the advent of chaos theory. For what corresponds to the linear approximation in the physical theory of dynamical systems would be the assumption of normally distributed errors in statistics. The normal distribution occupies, indeed, a privileged place in the theory of probability due to the central limit theorem which states that, under fairly broad conditions, in a process reflecting the operation of many independent factors, these will combine to yield an orderly behavior with deviations from the mean characterized by a Gaussian distribution. The classic case would be a laboratory experiment in which a given quantity is repeatedly measured; the errors about the mean value are typically normally distributed. Thus, for many phenomena the central limit theorem is a trustworthy guide.

The key point, however, is that for many other phenomena it fails altogether! What causes its downfall? The underlying assumption that fortuitous effects may be supposed to be independent. If there are, instead, correlations among them, as rule they will combine in such a fashion as to yield a probability distribution for gross quantities showing many more outliers than one would foresee on the basis of a normal distribution – what in current parlance one calls a fat tail. A good illustration of this effect, in physics, would be the fluid state of matter. There, unlike what happens in a gas (at low enough pressures, well away from the critical point) in which collisions among the constituent particles are comparatively infrequent, in a fluid the particles are jammed close enough together that collisions among them happen all the time and therefore cannot at all be neglected. Now, the effect of interparticle collisions is to introduce correlations, as is easy to see: pour a handful of ball bearings into a can and shake it. Then, the spheres will have to become correlated among themselves in the sense that, if two should happen to be located close by, this configuration tends to block a third sphere from entering the same location. In technical terms, one says that the point-two correlation function will exhibit excluded-volume effects. The characteristic signal, then, of a fluid is a two-point correlation function that falls off as a power law, much more slowly than exponential (as can be verified either directly by neutron diffraction or indirectly by numerical experiments).

Why fat tails matter in finance: the intervention of human actors means that collective effects can become paramount. When Nissam Taleb began his career as a public intellectual twenty years ago, awareness of this circumstance had not yet penetrated as deeply into the minds of economists and financiers as it should have. For instance, the celebrated Black-Scholes theory of option pricing is based on an assumption of thin tails. Failure to appreciate the crucial importance of fat tails has real-world consequences, too, for it can adversely affect decision-making at large financial institutions and contribute eventually to the need for bailouts (which is why we, as members of the general public should care; in a non-ideal or oligopolistic market one cannot simply rely on the traditional free-market mechanism of weeding out careless players due to bankruptcy – the ‘too big to fail’ syndrome). The author collects in the present work, Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications Papers and Commentary (STEM Press, 2020) several of his formal papers from the journal literature and affixes an introductory overview.

Taleb’s focus in the present work and our focus in the present review will be strictly on the data science itself, not any policy aspects (say, what if anything to do about the perverse incentive structure that drove the economic system in developed countries into the financial crisis of 2008). The first two chapters [pp. 1-18] consist in a not-particularly-useful glossary of terms, at least to the beginner who has yet to appreciate what they are all about. Parts I [pp. 19-124] and II [pp. 125-210], however, reprint a non-technical lecture Taleb once gave to the Darwin College and then several further chapters and appendices. These may be taken as typical of Taleb’s style: often polemical, but nicely illustrated with clear formulae and charts and free of cumbersome technical derivations (which are deferred to later parts of the book). In a qualitative sense, fat tails arise in situations where rare events play a disproportionate role in determining properties of the system. A rough measure of fat-tailedness of a distribution would be when one of the moments becomes infinite. Consequences of leaving the familiar zone: 1) the law of large numbers works more slowly, if at all; 2) there is a persistent small-sample effect on the estimated mean; 3) standard deviation and variance are not usable; 4) linear least-squared regression doesn’t work, but the maximum likelihood method can; 5) principal components analysis and factor analysis will likely produce spurious results [pp. 31-35]. Even large deviation theory fails to apply [p. 26]. Moreover, in Taleb’s view, Bayesian methods are of little help as well, as, for them, one needs a reliable prior, precisely what fat tails call into question [p. 56].

Thus, one encounters the problem in practice that dynamic hedging of financial options does not mitigate risk. How, in light of this troubling circumstance, Taleb proposes one ought to deal with fat tails (in the context of banking): pay attention not to the random variable itself but to the payoff or exposure [p. 57]. The first half of Part I contains an informal introduction while the second half gets more technical but not too formal (derivations and proofs postponed). A nice feature of Taleb’s exposition is the way he exemplifies his statements with diagrams and numerical examples – for instance, a statistical analysis of the S&P 500 [pp. 185ff] showing that its returns are power-law distributed with an asymmetry between upside and downside. Taleb also displays his practical bent by discussing ways of detecting ‘infinite’ moments in finite data sets: use the l^p norm for large p [p. 186]; and his view of the problem with econometrics in general [p. 199ff] –, namely, that these tools invite foolish risk taking [p. 200].

Taleb’s conclusion to Part II is worth reproducing here:

Many researchers invoke "outliers" or "peso problem" as acknowledging fat tails (or the role of the tails for the distribution), yet ignore them analytically (outside of Poisson models that are not possible to calibrate except after the fact: conventional Poisson jumps are thin-tailed). Our approach here is exactly the opposite: do not push outliers under the rug, rather build everything around them. In other words, just like the FAA and the FDA who deal with safety by focusing on catastrophe avoidance, we will throw away the ordinary under the rug and retain extremes as the sole sound approach to risk management. And this extends beyond safety since much of the analytics and policies that can be destroyed by tail events are inapplicable. [p. 204]

Part III concerns predictions, forecasting and uncertainty under fat-tail conditions. Without proper calibration, one is liable to get spurious results. The robust methods Taleb advocates tolerate a certain known inaccuracy for the sake of reliability in detecting large events, when they do occur (thus, on the principle that it is better to accept a quota of false positives in order to avoid too many false negatives). The second half of Part III consists in a paper on forecasting binary outcomes (such as an election). Here, there is a counter-intuitive effect in that the greater the uncertainty in the underlying security, the lower the volatility in the binary option. Taleb gives vent to his polemical tendency in criticizing what he sees as stark errors committed by political scientists and pollsters, in particular, in regard to the 2016 presidential election.

Part IV on inequality estimators under fat tails is again rather technical. It reprints two papers, one on estimating the Gini coefficient (a measure of the degree of inequality in the distribution of wealth across citizens of a country), which clearly falls under the rubric of fat tails in highly stratified societies such as, traditionally, Brazil or for that matter, ours in recent decades as a result of ongoing unchecked trends in late capitalism. The second paper concerns biases in the estimation of quantile contributions. After reading the general introduction discussed above, only the expert will be left unsatisfied and want to study these papers so as to learn about what robust methods there might be to do as well as one can under the circumstances (a circumstance often encountered by practitioners of data science who cannot afford the luxury of remaining in the land of pure theory but are assigned the task of coming up with something defensible to report on a real data set). Nevertheless, one may extract an occasional comment of wider interest, such as the following:

The mistake appears to be commonly made in common inference about fat-tailed data in the literature. The very methodology of using concentration and changes in concentration is highly questionable. For instance, in the thesis by Steven Pinker that the world is becoming less violent, we note a fallacious inference about the concentration of damage from wars from an estimator with minutely small population in relation to the fat-tailedness. Owing to the fat-tailedness of war casualties and consequences of violent conflicts, an adjustment would rapidly invalidate such claims that violence from war has statistically experienced a decline. [p. 283]

Part VI on meta-probability promises to be more interesting conceptually than what has gone before. In chapter 17, Taleb discusses how thick tails emerge from recursive epistemic uncertainty, or the opposite of the central limit theorem; in chapter 19, the meta-distribution of p-values and p-value hacking. The implication is that CAPM measures of risk are unrealistic. Not only stationary models such as CAPM are affected, though; Taleb contends here also that dynamic hedging is a mathematical impossibility. What is one to do about this, in practice? See chapter 23 on option pricing under power laws: a robust heuristic, and chapter 25 on tail risk constraints and maximum entropy, on maximum entropy methods for portfolio selection where one is concerned especially with minimizing downside risk.

This concludes our overview of contents. To compare the present work with non-technical writings by Nassim Taleb, such as his Black Swan (2007, which we intend to review in a moment), here, in the context of fairly technical discussions in probability theory, he keeps himself sedate and detached, whereas in his popular works there is this tension: rhetorically, Taleb wants to exaggerate the difference between thin and fat tails to the point that one would suppose no theoretical understanding of the latter is possible at all. Of course, by fiat one can represent a black swan as an event so large and disruptive as to defy any theoretical understanding, yet there are numerous intermediate regimes that are of theoretical interest (e.g., power laws in molecular hydrodynamics, Langevin equation, Lévy flights etc.). As a would-be guru, Taleb presses the case for black swans as far as it can go in order to support a world-view skeptical of the very possibility of nomothetic comprehension. A strategy such as this turns out to be self-defeating, in that it forces one to disregard large swathes of experience where a degree of law-like regularity prevails (if not precisely the Newtonian ideal of clockwork mechanism) so as to declare victory by definition, as it were. The fact of the matter is that our world displays all types of qualitative behavior, from regimes characterized by thin tails to intermediate cases and on to those (relatively few, but important in human affairs) having thick tails.

Let us conclude with an observation this reviewer once heard from the complexity theorist Scott Page: randomness due to complexity is different in kind from randomness due to what one is accustomed to think of as chance occurrences in typical physical systems. For, in the latter, random fluctuations are weak, as rule, but still exist and can add up to cause unexpected effects (as an illustration, weak correlated noise appears to be behind the phenomenon of rogue waves in the ocean). In such cases, the power spectrum of events will be found not to depart greatly from the usual; i.e., although outliers occur with a certain (possibility enhanced, relative to naïve expectation) frequency, one is still in a thin-tailed regime. Complex systems such as tend to arise spontaneously in human societies, on the other hand, obey an entirely different regime of qualitative behavior: collective effects dominate and one has to pay attention to cooperative resp. anti-cooperative interactions among intelligent agents who can act on global information. As a result, the system is always unstable to the emergence of large deviations, which, when they do occur, propagate rapidly. In other words, events will follow a fat-tailed distribution, in which some account must be made for what Taleb has taken to calling his ‘black swans’.

Thus, one must receive Taleb’s drastic warnings with a grain of salt: they do not mean that the world has suddenly become incomprehensible, only that certain situations encountered particularly in the domain of economics and finance do tend to produce the occasional large deviation and that, therefore, the risks we must be prepared to face are of a different nature than one might suppose based on unschooled intuition. This book shows off Taleb at his best, in the area of his competence, while his popular writings venture into inflammatory metaphysical speculations outside his purview. Here, he offers in more detached terms many practical suggestions for data scientists who have to deal with a messy real world in which fat tails sometimes matter.

[Brahm · Rating 3 out of 5]

I read maybe the first 70 pages and skimmed the rest to try and take in the graphs, commentary, and footnotes. I did not make an effort to understand most of the math and derivations, but what I did consume helped broaden my understanding of concepts from his more human-readable books.

[Alexej Gerstmaier · Rating 5 out of 5]

Most of the math went over my head (it's the *technical* Incerto after all) but the non math content alone is worth it.

Might re-read after acquiring more stats/math/quant knowledge

[Mark Mitchell · Rating 5 out of 5]

The "Technical Incerto" is the mathematical companion to the series of books beginning with Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets and which includes both The Black Swan: The Impact of the Highly Improbable and Antifragile: Things That Gain from Disorder. While Taleb's other books are written without equations, and make mathematical arguments using intuitive, non-technical presentations, Statistical Consequences of Fat Tails: Real World Preasymptotics, Epistemology, and Applications is thoroughly mathematical.

For those who have read and enjoyed Taleb's other books and who have a mathematical background, the Technical Incerto will be deeply satisfying. This book justifies some of Taleb's philosophical positions in rigorous terms, including those relating to the importance of "power law" or "fat-tailed" distributions in the real world. Moreover, Taleb's mathematical arguments make explicit the assumptions that he makes in his more popular writing, which helps the critical reader understand his arguments.

The book will be incomprehensible to those uncomfortable with mathematics and will be best appreciated by those with extensive knowledge of probability and statistics. Taleb presumes comfort with calculus, continuous probability distributions, characteristic functions, stochastic processes, and other such topics. The presentation is at the level of a graduate-school seminar or conference paper; readers who wish to get the most from the book should expect to work through derivations and proofs, re-discovering steps that Taleb has omitted from his proofs. While there are errors in some of the proofs, there are none (to the best of my knowledge) in the significant conclusions; most of the errors are typographical or minor in nature. Nevertheless, the careful reader may find herself confused until she determines what Taleb intended.

[Zach Mackin · Rating 5 out of 5]

One of the best textbooks there is, and a completely underrepresented field. Definitely a book I will keep coming back to

[Artūrs Kaņepājs · Rating 4 out of 5]

"Probability is just a kernel inside an integral or a summation, not a real thing on its own. The economic world is about quantitative payoffs." With this quote, probability for me went in the bucket of sometimes useful but false stuff (like religion, materialism, speciesism).

I'm working in quantitative risk management in banking. Many know Taleb's despisal towards VaR. It was encouraging that he points at potential improvements, like EVT for cVaR. But given the stakeholders and competences in the industry I don't think these will be taken on. We are quite stuck with a number of procedures and measures that try to compensate for the lack of statistical rigor.

Some other key takeaways:
- modern portfolio theory is invalid as Gaussians are assumed everywhere
- MAD is a more robust measure than STD
- under large uncertainty, implied probablilities for binary payoffs are close to 50-50 until the very end
- Brier score sucks in many contexts
- Wealth may show up as less equal than income just b/c of passage of time, as wealth accumulation is a fat tailed cumulative process.

The design of the book was excellent and the text - engaging. The reasons for 4 stars:
- The abundance of typos.
- Many "research chapters" consist entirely of already published material available elsewhere - they fit in well, though obviously were not optimized to fit in perfectly at the time of writing.

[Enrique · Rating 2 out of 5]

Mismo problema que con Dynamic Hedging: para alguien absolutamente nulo en matemáticas como yo, salvo picotear unos cuantos excelentes razonamientos filosóficos, poco podrá sacar en claro. Alguien con mayor dominio matemático, en cambio, probablemente le dé 4-5 estrellas y encuentre su parque de atracciones de placer matemático entre sus fórmulas y parámetros.

[Franklin Fuchs · Rating 5 out of 5]

Fantastic read and incredibly pragmatic. Loved the technical deep dive equivalent to Taleb’s incerto series. If/When the second volume of the technical incerto comes out, it will be an instant read for me.

[Alex Bilzerian · Rating 5 out of 5]

Indispensable for practitioners.

[Jaime Gacitua · Rating 3 out of 5]

I read the first 5-6 chapters, skimming over the math.
I flipped through the rest of the book.

Very interesting to learn fat tails are everywhere.
And the consequences of that.
And the mistakes people have made for overlooking them.

I am starting to sense the value of the "outlier data," which I've discarded countless times.
Next time I see data science results, I wonder if the underlying distribution is thin or fat-tailed.

I saw typos, and the book does not seem well organized as a unified storyline.
It's a collection of manuscripts.

[Ron Me]

This is the one we've been waiting for, kids! It touches on a wide variety of issues that have bothered me over the years. Gets "Math Book of the Year" award for my purposes! Not for the mathematically challenged.

N.B.: Preprint is ArXiv 2001.10488.

[John · Rating 4 out of 5]

Good technical detail in here, a great reference on statistics.

[Jerry Luan · Rating 4 out of 5]

It's a very tough read and unnecessary for most people. This book is the mathematical presentation of NNT's ideas in Incerto.

[Matteo Pedretti · Rating 1 out of 5]

chi me l'ha fatto fare...

[José González · Rating 4 out of 5]

Great potential for this book. Still in the developing phase, so can't fully appreciate it.

[Ferhat Culfaz · Rating 3 out of 5]

Very rich book but very technical and not for everyone. A good reference source.

[Alberto · Rating 1 out of 5]

to pick up in the future