# Nassim Nicholas Taleb's research while affiliated with City University of New York - Brooklyn College and other places

## Publications (98)

Preprint
Full-text available
We extend techniques and learnings about the stochastic properties of nonlinear responses from finance to medicine, particularly oncology where it can inform dosing and intervention. We define antifragility. We propose uses of risk analysis to medical problems, through the properties of nonlinear responses (convex or concave). We 1) link the convex...
Article
Full-text available
Background: Pre-pandemic empirical studies have produced mixed statistical results on the effectiveness of masks against respiratory viruses, leading to confusion that may have contributed to organizations such as the WHO and CDC initially not recommending that the general public wear masks during the COVID-19 pandemic. Methods: A threshold-based d...
Article
A technology should be judged in how it solves recognized problems, not by its technical appeal
Preprint
Full-text available
This discussion applies quantitative finance methods and economic arguments to cryptocurrencies in general and bitcoin in particular -- as there are about $10,000$ cryptocurrencies, we focus (unless otherwise specified) on the most discussed crypto of those that claim to hew to the original protocol (Nakamoto 2009) and the one with, by far, the lar...
Preprint
Full-text available
Face masks have been widely used as a protective measure against COVID-19. However, pre-pandemic experimental studies have produced mixed results regarding their effectiveness against respiratory viruses, leading to confusion over whether masks protect the wearer, or only those with whom the wearer interacts. Such confusion may have contributed to...
Article
We discuss common errors and fallacies when using naive “evidence based” empiricism and point forecasts for fat-tailed variables, as well as the insufficiency of using naive first-order scientific methods for tail risk management. We use the COVID-19 pandemic as the background for the discussion and as an example of a phenomenon characterized by a...
Preprint
We discuss common errors and fallacies when using naive "evidence based" empiricism and point forecasts for fat-tailed variables, as well as the insufficiency of using naive first-order scientific methods for tail risk management. We use the COVID-19 pandemic as the background for the discussion and as an example of a phenomenon characterized by a...
Article
Full-text available
Models can help us determine how to stop the spread of COVID-19, but it is important to distinguish between that which models can and cannot predict. All models’ assumptions fail to describe the details of most real-world systems, but these systems may possess large-scale behaviors that do not depend on all these details. A simple model that correc...
Article
The COVID-19 pandemic has been a sobering reminder of the extensive damage brought about by epidemics, phenomena that play a vivid role in our collective memory, and that have long been identified as significant sources of risk for humanity. The use of increasingly sophisticated mathematical and computational models for the spreading and the implic...
Preprint
Applying a modification of Extreme value Theory (thanks to a dual distribution technique by the authors on data over the past 2,500 years, we show that pandemics are extremely fat-tailed in terms of fatalities, with a marked potentially existential risk for humanity. Such a macro property should invite the use of Extreme Value Theory (EVT) rather t...
Preprint
Empirical distributions have their in-sample maxima as natural censoring. We look at the "hidden tail", that is, the part of the distribution in excess of the maximum for a sample size of $n$. Using extreme value theory, we examine the properties of the hidden tail and calculate its moments of order $p$. The method is useful in showing how large a...
Article
We map the difference between (univariate) binary predictions, bets and “beliefs” (expressed as a specific “event” will happen/will not happen) and real-world continuous payoffs (numerical benefits/harm from an event) and show the effect of their conflation and mischaracterization in the decision-science literature. We also examine the differences...
Preprint
This is an epistemological approach to errors in both inference and risk management, leading to necessary structural properties for the probability distribution. Many mechanisms have been used to show the emergence of fat tails. Here we follow an alternative route, the epistemological one, using counterfactual analysis, and show how nested uncertai...
Preprint
We build a heuristic that takes a given option price in the tails with strike K and extends (for calls, all strikes > K, for put all strikes < K) assuming the continuation falls into what we define as "Karamata Constant" over which the strong Pareto law holds. The heuristic produces relative prices for options, with for sole parameter the tail inde...
Preprint
What do binary (or probabilistic) forecasting abilities have to do with overall performance? We map the difference between (univariate) binary predictions, bets and "beliefs" (expressed as a specific "event" will happen/will not happen) and real-world continuous payoffs (numerical benefits or harm from an event) and show the effect of their conflat...
Article
Article
This paper presents an operational metric for univariate unimodal probability distributions with finite first moments in [0,1], where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. It is based on the question, “how much data does one need to make meaningful statements about a given dataset?” Applications: Among others, it •hel...
Preprint
This paper applies risk analysis to medical problems, through the properties of nonlinear responses (convex or concave). It shows 1) necessary relations between the nonlinearity of dose-response and the statistical properties of the outcomes, particularly the effect of the variance (i.e., the expected frequency of the various results and other prop...
Chapter
This paper applies risk analysis to medical problems, through the properties of nonlinear responses (convex or concave). It shows (1) necessary relations between the nonlinearity of dose-response and the statistical properties of the outcomes, particularly the effect of the variance (i.e., the expected frequency of the various results and other pro...
Chapter
We propose an approach to compute the conditional moments of fat-tailed phenomena that, only looking at data, could be mistakenly considered as having infinite mean. This type of problems manifests itself when a random variable Y has a heavy-tailed distribution with an extremely wide yet bounded support.
Article
This note presents an operational measure of fat-tailedness for univariate probability distributions, in $[0,1]$ where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. Among others,1) it helps assess the sample size needed to establish a comparative $n$ needed for statistical significance, 2) allows practical comparisons across...
Article
We study the problems related to the estimation of the Gini index in presence of a fat-tailed data generating process, i.e. one in the stable distribution class with finite mean but infinite variance (i.e. with tail index α∈(1,2)). We show that, in such a case, the Gini coefficient cannot be reliably estimated using conventional nonparametric metho...
Article
The more volatile the prediction the closer to an even call
Article
Under infinite variance, the Gini coefficient cannot be reliably estimated using conventional nonparametric methods. We study different approaches to the estimation of the Gini index in presence of a heavy tailed data generating process, that is, one with Paretan tails and/or in the stable distribution class with finite mean but non-finite variance...
Article
We consider the estimation of binary election outcomes as martingales and propose an arbitrage pricing when one continuously updates estimates. We argue that the estimator needs to be priced as a binary option as the arbitrage valuation minimizes the conventionally used Brier score for tracking the accuracy of probability assessors. We create a dua...
Article
We examine random variables in the power law/slowly varying class with stochastic tail exponent, the exponent $\alpha$ having its own distribution. We show the effect of stochasticity of $\alpha$ on the expectation and higher moments of the random variable. For instance, the moments of a right-tailed or right-asymmetric variable, when finite, incre...
Article
Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be considered when dealing with tail-risk assessment...
Article
Pasquale Cirillo and Nassim Nicholas Taleb respond to December's “Ask a Statistician” column, explaining more about their work to understand the risk of violent conflicts
Article
We present an exact probability distribution (meta-distribution) for p-values across ensembles of statistically identical phenomena, as well as the distribution of the minimum p-value among $m$ independents tests. We derive the distribution for small samples $2<n \leq n^*\approx 30$ as well as the limiting one as the sample size $n$ becomes large....
Article
We examine statistical pictures of violent conflicts over the last 2000 years, providing techniques for dealing with the unreliability of historical data. We make use of a novel approach to deal with fat-tailed random variables with a remote but nonetheless finite upper bound, by defining a corresponding unbounded dual distribution (given that pote...
Article
We propose a methodology to look at violence in particular, and other aspects of quantitative historiography in general, in a way compatible with statistical inference, which needs to accommodate the fat-tailedness of the data and the unreliability of the reports of conflicts. We investigate the theses of “long peace” and drop in violence and find...
Chapter
Standard economic theory makes an allowance for the agency problem, but not the compounding of moral hazard in the presence of informational opacity, particularly in what concerns high-impact events in fat tailed domains (under slow convergence for the law of large numbers). Nor did it look at exposure as a filter that removes nefarious risk takers...
Article
We propose an approach to compute the conditional moments of fat-tailed phenomena that, only looking at data, could be mistakenly considered as having infinite mean. This type of problems manifests itself when a random variable Y has a heavy- tailed distribution with an extremely wide yet bounded support. We introduce the concept of dual distributi...
Article
Direct measurements of Gini coefficients by conventional arithmetic calculations are a poor estimator, even if paradoxically, they include the entire population, as because of super-additivity they cannot lend themselves to comparisons between units of different size, and intertemporal analyses are vitiated by the population changes. The Gini of ag...
Article
Portfolio selection in the financial literature has essentially been analyzed under two central assumptions: full knowledge of the joint probability distribution of the returns of the securities that will comprise the target portfolio; and investors’ preferences are expressed through a utility function. In the real world, operators build portfolios...
Article
We examine all possible statistical pictures of violent conflicts over common era history with a focus on dealing with incompleteness and unreliability of data. We apply methods from extreme value theory on log-transformed data to remove compact support, then, owing to the boundedness of maximum casualties, retransform the data and derive expected...
Article
Proof that under simple assumptions, such as constraints of Put-Call Parity, the probability measure for the valuation of a European option has the mean derived from the forward price which can, but does not have to be the risk-neutral one, under any general probability distribution, bypassing the Black-Scholes-Merton dynamic hedging argument, and...
Article
Sample measures of top centile contributions to the total (concentration) are downward biased, unstable estimators, extremely sensitive to both sample and population size and concave in accounting for large deviations. It makes them particularly unfit in domains with power law tails, especially for low values of the exponent. These estimators can v...
Article
Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be considered when dealing with tail-risk assessment...
Article
We examine statistical pictures of violent conflicts over the last 2000 years, finding techniques for dealing with incompleteness and unreliability of historical data. We introduce a novel approach to apply extreme value theory to fat-tailed variables that have a remote, but nonetheless finite upper bound, by defining a corresponding unbounded dua...
Article
Full-text available
In the world of modern financial theory, portfolio construction has traditionally operated under at least one of two central assumptions: the constraints are derived from a utility function and/or the multivariate probability distribution of the underlying asset returns is fully known. In practice, both the performance criteria and the informationa...
Article
Full-text available
We present a non-naive version of the Precautionary Principle (PP) that allows us to avoid paranoia and paralysis by confining precaution to specific domains and problems. PP is intended to deal with uncertainty and risk in cases where the absence of evidence and the incompleteness of scientific knowledge carries profound implications and in the pr...
Article
Full-text available
In fat-tailed domains, sample measures of top centile contributions to the total (concentration) are biased, unstable estimators extremely sensitive to sample size and concave in accounting for large deviations. They can vary over time merely from the increase of sample space, thus providing the illusion of structural changes in concentration. They...
Article
Full-text available
Religions come with risk-managing interdicts and heuristics, and they carry such interdicts and heuristics across generations. We remark on such facets of religion in relation to a propensity among some decision scientists and others to regard practices that they cannot understand as being irrational, biased, and so on.
Article
Proof that under simple assumptions, such as con- straints of Put-Call Parity, the probability measure for the valuation of a European option has the mean of the risk-neutral one, under any general probability distribution, bypassing the Black-Scholes-Merton dynamic hedging argument, and without the requirement of complete markets. We confirm that...
Article
For more than a century, the economics profession has extended its reach to encompass policy formation and institutional design while largely ignoring the ethical challenges that attend the profession’s influence over the lives of others. Economists have proved to be disinterested in ethics, which, embracing emotivism, they often treat as a matter...
Article
Using Jeff Holman's comments in Quantitative Finance to illustrate 4 critical errors students should learn to avoid: 1) Mistaking tails (4th moment) for volatility (2nd moment), 2) Missing Jensen's Inequality, 3) Analyzing the hedging wihout the underlying, 4) The necessity of a numeraire in finance.
Article
The full-length book provides a mathematical framework for decision making and the analysis of (consequential) hidden risks, those tail events undetected or improperly detected by statistical machinery; and substitutes fragility as a more reliable measure of exposure. Model error is mapped as risk, even tail risk.Risks are seen in tail events rathe...
Article
Full-text available
Standard economic theory makes an allowance for the agency problem, but not the compounding of moral hazard in the presence of informational opacity, particularly in what concerns high-impact events in fat tailed domains. Nor did it look at exposure as a filter that removes bad risk takers from the system so they stop harming others. But the ancien...
Article
The literature of heavy tails (typically) starts with a random walk and finds mechanisms that lead to fat tails under aggregation. We follow the inverse route and show how starting with fat tails we get to thin-tails when deriving the probability distribution of the response to a random variable. We introduce a general dose-response curve and argue...
Article
Owing to the convexity of the payoff of out-of-the money options, an extremely small probability of a large deviation unseen in past data justifies rationally buying them, or at least justifies excessive caution in not being exposed to them, particularly those options that are extremely nonlinear in response to market movement or changes in implied...
Article
There are serious differences between predictions, bets, and exposures that have a yes/no type of payoff, the "binaries", and those that have varying payoffs, which we call the "vanilla". Real world exposures tend to belong to the vanilla category, and are poorly captured by binaries. Vanilla exposures are sensitive to Black Swan effects, model err...
Article
Standard economic theory makes an allowance for the agency problem, but not the compounding of moral hazard in the presence of informational opacity, particularly in what concerns high-impact events in fat tailed domains. Nor did it look at exposure as an evolutionary filter that removes bad risk takers from the system so they stop harming others....
Article
In the presence of a layer of metaprobabilities (from metadistribution of the parameters), the asymptotic tail exponent corresponds to the lowest possible tail exponent regardless of its probability. The problem explains “Black Swan ” effects, i.e., why measurements tend to chronically underestimate tail contributions, rather than merely deliver im...
Article
Full-text available
We provide a mathematical definition of fragility and antifragility as negative or positive sensitivity to a semi-measure of dispersion and volatility (a variant of negative or positive "vega") and examine the link to nonlinear effects. We integrate model error (and biases) into the fragile or antifragile context. Unlike risk, which is linked to ps...
Article
Full-text available
This paper presents a simple heuristic measure of tail risk, which is applied to individual bank stress tests and to public debt. Stress testing can be seen as a first order test of the level of potential negative outcomes in response to tail shocks. However, the results of stress testing can be misleading in the presence of model error and the unc...
Article
It is assumed that while portfolio theory fails with daily returns, that it would work with yearly returns, an standard argument recently repeated in Treynor (2011). This paper debunks the confusion that daily returns, when non-Gaussian but with finite variance can aggregate to thin tails. Alas, portfolio theory fails in both the short and the long...
Article
This article argues that the crisis of 2007–2008 happened because of an explosive combination of agency problems, moral hazard, and “scientism.” The authors analyze the varied behaviors, ideas and effects that in combination created a financial meltdown, and discuss the players responsible for the consequences. In formulating a set of expectations...
Article
Where the problem is not expert underestimation of randomness, but more: the tools themselves used in regression analyses and similar methods underestimate fat tails, hence the randomness in the data. We should avoid imparting psychological explanations to errors in the use of statistical methods.
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The main results are 1) definition of fragility, antifragility and model error (and biases) from missed nonlinearities and 2) detection of these using a single "fast-and-frugal", model-free, probability free heuristic. We provide an expression of fragility and antifragility as negative or positive sensitivity to second order effects, i.e., dispersi...
Article
Ex ante predicted outcomes should be interpreted as counterfactuals (potential histories), with errors as the spread between outcomes. But error rates have error rates. We reapply measurements of uncertainty about the estimation errors of the estimation errors of an estimation treated as branching counterfactuals. Such recursions of epistemic uncer...
Article
Full-text available
Using the metadistribution of possible distributions for a given measure, we define a condition under which it is possible to make a decision based on the observation of random variable, which we call "statistical decidability". We provide a sufficient condition on the metadistribution for the decision to be "statistically decidable" and conjecture...
Article
This discussion makes the distinction inside the Fourth Quadrant "Black Swan Domain" between fragile and robust to model (or representational) error on the basis of convexity. The notion of model error as a convex or concave stochastic variable; why deficit forecasting errors are biased in one direction; why large is fragile to errors; how economic...
Article
This paper establishes the case for a fallacy of economies of scale in large aggregate institutions and the effects of scale risks. The problem of rogue trading and excessive risk taking is taken as a case example. Assuming (conservatively) that a firm exposure and losses are limited to its capital while external losses are unbounded, we establish...
Article
This paper - while a standalone invited essay written for a special crisis issue of New Political Economy - synthesizes the various technical documents by the author as related to the financial crisis. It can also be used as a technical companion to The Black Swan (2007-2010).
Chapter
IntroductionClasses of UncertaintyOur ArgumentPredicting Other People's ActionPredicting One's Own ActionsPredicting Group ActionThe Danger of PredictionReferences
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This paper examines the risk externalities stemming from the size of institutions. Assuming (conservatively) that a firm risk exposure is limited to its capital while its external (and random) losses are unbounded we establish a condition for a firm to be too big to fail. In particular, expected risk externalities’ losses conditions for positive fi...
Article
The point of The Black Swan is that both empirical knowledge (i.e. extrapolating statistics) and a priori theories fail in the tails and it is vital to "robustify" against it using the concepts of "the fourth quadrant". The point has been garbled by members of the economics establishment that claim mistakenly "we know that" and "we know about fat t...
Article
This conclusion aims to summarize the major issues surrounding forecasting, as well as the extensive empirical evidence proving our inability to accurately predict the future. In addition, it discusses our resistance to accepting such inaccurate predictions, while putting forwards a number of ideas aimed at a complex world where accurate forecastin...
Article
Full-text available
This special section aims to demonstrate the limited predictability and high level of uncertainty in practically all important areas of our lives, and the implications of this. It summarizes the huge body of solid empirical evidence accumulated over the past several decades that proves the disastrous consequences of inaccurate forecasts in areas ra...
Article
The paper presents evidence that econometric techniques based on variance-L2 norm-are flawed and do not replicate. The result is un-computability of the role of tail events. The paper proposes a methodology to calibrate decisions to the degree (and computability) of forecast error. It classifies decision payoffs in two types: simple (true/false or...
Article
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Black Swan events are almost impossible to predict. Instead of perpetuating the illusion that we can anticipate the future, risk management should try to reduce the impact of the threats we don't understand.
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Large institutions are disproportionately more fragile to Black Swans. This paper establishes the case for a fallacy of economies of scale in large aggregate institutions. The problem of rogue trading is taken as a case example of hidden risks where rogue traders and losses are considered independently and dependently of the institution’s size. Bot...
Article
Full-text available
Option traders use a heuristically derived pricing formula which they adapt by fudging and changing the tails and skewness by varying one parameter, the standard deviation of a Gaussian. Such formula is popularly called “Black–Scholes–Merton” owing to an attributed eponymous discovery (though changing the standard deviation parameter is in contradi...
Article
Outside the Platonic world of financial models, assuming the underlying distribution is a scalable power law, we are unable to find a consequential difference between finite and infinite variance models - a central distinction emphasized in the econophysics literature and the financial economics tradition. While distributions with power law tail ex...
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Full-text available
Finance professionals, despite regular exposure to notions of volatility, seem to confuse mean absolute deviation with standard deviation. In some fat tailed markets, theoretical Gaussian variables can be underestimated by as much as 90%. It is not a lack of statistical knowledge that appears to be the impediment, but rather difficulty in translati...
Article
Full-text available
Finance professionals, who are regularly exposed to notions of volatility, seem to confuse mean absolute deviation with standard deviation, causing an underestimation of 25% with theoretical Gaussian variables. In some fat tailed markets the underestimation can be up to 90%. The mental substitution of the two measures is consequential for decision...
Article
Options traders use a pricing formula which they adapt by fudging and changing the tails and skewness by varying one parameter, the standard deviation of a Gaussian. Such formula is popularly called "Black-Scholes-Merton" owing to an attributed eponymous discovery (though changing the standard deviation parameter is in contradiction with it). Howev...
Article
Full-text available
While modern financial theory holds that options values are derived by dynamic replication, they can be correctly valued far more simply by long familiar static and actuarial arguments that combine stochastic price evolution with the no-arbitrage relation between cash and forward contracts.
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This paper surveys the behavioral literature in search of possible explanations for the preference for negative skewness on the part of economic agents. 1) We relate the mathematical properties of skewness to biases in inductive inference and suggest further research needed for the conditions of the real world in which agents are not presented the...
Article
The generator of a random process is seldom observable in the social sciences, only its realizations, which may or may not reveal its nature. Yet knowledge about the generator is what is needed to infer the true moment properties of the process (expectation, variance) . Distinguis hing between Knightian risk and Knightian uncertainty becomes imposs...
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We thank participants at the American Association of Artificial Intelligence Symposium on Chance Discovery in Cape Cod in November 2002, Stanford University Mathematics Seminar in March 2003, the Italian Institute of Risk Studies in April 2003, and the ICBI Derivatives conference in Barcelona in May 2003. We thank Benoit Mandelbrot and Didier Sorne...

## Citations

... In the 'rank bargaining' treatment, individuals were informed about their ranks in the test. Although the ranks were randomly assigned, subjects were told that they were assigned on the basis of their performances in the test (Rousu et al. 2015;Ball et al. 2001;DeMartino and McCloskey 2016). Higher-ranked subjects were matched with lower-ranked subjects for the bargaining experiment. ...
... • The presence of concavity in the tails of the distribution implies a silent risk. This approach was used in stress testing by the International Monetary Fund (IMF) where the degree of concavity in the tail was used as an indicator of the severity of tail exposure, see Taleb, Canetti et al. [102]. ...
... o "Conclusions: "There is currently no evidence from RCTs demonstrating that the use of cloth or medical masks prevents the transmission of SARSCoV-2 in the community setting" (Chetty et al., 2021) o Despite the common sense evidence pertaining to masks, as I have indicated previously, papers that seek to minimize or potentially discredit the lack of real impact of masks in real life continue to be published [truly, there is nothing wrong with debate or disagreementsif it allows for a bidirectional flow of arguments and is an effort to seek the truth, but that has not been the case]. Kollepara et al. (2021) state: ...
... Un metaverso es un universo digital de cierta especificidad temática donde conviven narrativas individuales en una interfaz estética y muchas veces jugable cuya ambición es ser mímesis de la realidad en la medida en la que la reproduzca ya no buscando ser idéntica, sino amplificando todas sus posibilidades interactivas (Taleb, 2021). Un gran paradigma universal que atrape la individualidad en el gran bazar hiperdigital, mercadeando a través de microtransacciones oficiales, dones no oficiales obsequiados en salas de chat y prótesis de avatar. ...
... Such scenarios are problematic due to the high levels of uncertainty of long-term (multiple weeks and months) case number forecasts; a generic feature of mathematical epidemiological models which has been put under the spotlight by the COVID-19 pandemic 40,42,44,45 . Based on our results, we argue that a main benefit of epidemiological models comes from their use as shortterm monitoring systems. ...
... In an attempt to obtain better predictions, it may be tempting to include more details and fine-tune the model assumptions. However, arbitrarily focusing on some assumptions and details while losing sight of others is counterproductive [12]. Which details are relevant depends on the question of interest; the inclusion or exclusion of details in a model must be justified depending on the modeling objectives. ...
... Taleb (2020) notes the tail exponent of a power law function captures (by extrapolation) the low-probability deviation not seen in the data and plays an important role in determining the mean. Moreover, Cirillo and Taleb (2020) show that the use of naïve statistics, such as the sample mean, may dramatically underestimate risk. Importantly, the lower the economic magnitude of the exponent, the higher is the impact of those low-probability deviations not seen in the empirical data. ...
... Modern decision theory has overcome the monoculture of rational choice theory long ago: Since the inception of the subfield in the 1950s (Simon 1955), behavioral economists have been working out a laundry list of biases and heuristics, explaining behavioral anomalies and sawing away at the hegemony of the mathematically convenient but unrealistic rational choice theory (Kahneman andTversky 1979, Thaler andShefrin 1981). Proponents of this stream of the literature have since received both Nobel Prizes and considerable pushback with regards to the real-world generalizability of their results (Gigerenzer 1991, Gigerenzer and Brighton 2009, Taleb 2020). Callahan and Elliott (1996) mark an early -if not much-noticed -call to integrate narrative into behavioral economics and thus put the content of human deliberation into focus in addition to its processes and outcomes. ...
... off updating today's prediction to match our expectation of tomorrow's forecast. In other words, such sequence of probabilistic predictions should be a martingale [Augenblick and Rabin, 2021, Foster and Stine, 2021, Gelman et al., 2020, Taleb et al., 2019, Taleb, 2018, Ely et al., 2015. But aside from satisfying this martingale property, the forecast trajectories can vary widely from one setting to the next. ...
... As for PIs, Gaba et al. [191] outlined six of such heuristics, namely, (1) the simple average, (2) the median, (3) and envelope approach, (4) an interior trimming method, (5) an exterior trimming method, and (6) probability averaging of endpoints. These heuristics were subsequently promoted by Grushka-Cockayne and Jose [192] to post-process the PIs submitted during the M4 forecasting competition [193][194][195]. With little surprise, interval aggregation was found to improve the calibration and accuracy. ...