The Foundations of Probability with Black Swans
We extend the foundation of probability in samples with rare events that are potentially catastrophic, called black 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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2010-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2010/838240 |
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| Summary: | We extend the foundation of probability in samples with rare
events that are potentially catastrophic, called black 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). |
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| ISSN: | 1687-952X 1687-9538 |