The Foundations of Probability with Black Swans

Graciela Chichilnisky

doi:10.1155/2010/838240

Abstract

We extend the foundation of probability in samples with rare events that are potentially catastrophic, calledblack swans, such as natural hazards, market crashes, catastrophic climate change, and species extinction. Such events are generally treated as ‘‘outliers’’ and disregarded. We propose a new axiomatization of probability requiring equal treatment in the measurement of rare and frequent events—the Swan Axiom—and characterize the subjective probabilities that the axioms imply: these are neither finitely additive nor countably additive but a combination of both. They exclude countably additive probabilities as in De Groot (1970) and Arrow (1971) and are a strict subset of Savage (1954) probabilities that are finitely additive measures. Our subjective probabilities are standard distributions when the sample has no black swans. The finitely additive part assigns however more weight to rare events than do standard distributions and in that sense explains the persistent observation of ‘‘power laws’’ and ‘‘heavy tails’’ that eludes classic theory. The axioms extend earlier work by Chichilnisky (1996, 2000, 2002, 2009) to encompass the foundation of subjective probability and axiomatic treatments of subjective probability by Villegas (1964), De Groot (1963), Dubins and Savage (1965), Dubins (1975) Purves and Sudderth (1976) and of choice under uncertainty by Arrow (1971).

Highlights

Black swans are rare events with important consequences, such as market crashes, natural hazards, global warming, and major episodes of extinction
We show that countably additive measures are insensitive to black swans: they assign negligible weight to rare events, no matter how important these may be, treating catastrophes as outliers
Each approach has advantages and shortcomings. It seems that the approach offered here may be superior on four counts: i it retains linearity of probabilities, ii it identifies Monotone Continuity as the reason for underestimating the measurement of catastrophic events, an axiom that depends on a technical definition of continuity and has no other compelling feature, iii it seems easier to explain and to grasp, and iv it may be easier to use in applications

Summary

Introduction

Black swans are rare events with important consequences, such as market crashes, natural hazards, global warming, and major episodes of extinction. It defines a new type of probabilities that coincide with standard distributions when the sample is populated only by relatively frequent events. They are a mixture of Journal of Probability and Statistics countable and finitely additive measures, assigning more weight to black swans than do normal distributions, and predicting more realistically the incidence of “outliers,” “power laws,” and “heavy tails” 1, 2. The new results provided here show that the standard axiom of decision theory, Monotone Continuity, is equivalent to De Groot’s Axiom SP4 that lies at the foundation of classic likelihood theory Proposition 2.1 and that both of these axioms underestimate rare events no matter how catastrophic they may be. It seems that the approach offered here may be superior on four counts: i it retains linearity of probabilities, ii it identifies Monotone Continuity as the reason for underestimating the measurement of catastrophic events, an axiom that depends on a technical definition of continuity and has no other compelling feature, iii it seems easier to explain and to grasp, and iv it may be easier to use in applications

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The Axiom of Choice