Exchangeability and sets of desirable gambles

Gert De Cooman; Erik Quaeghebeur

doi:10.1016/j.ijar.2010.12.002

Exchangeability and sets of desirable gambles

Gert De Cooman
, Erik Quaeghebeur

Research output: Contribution to journal › Article › Academic › peer-review

62 Citations (Scopus)

Abstract

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti's Finite and Infinite Representation Theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.

Original language	English
Pages (from-to)	363-395
Number of pages	33
Journal	International Journal of Approximate Reasoning
Volume	53
Issue number	3
DOIs	https://doi.org/10.1016/j.ijar.2010.12.002
Publication status	Published - Apr 2012
Externally published	Yes

Funding

The authors thank Teddy Seidenfeld for sharing some of the things he knows about Henry Kyburg and his work; they were both interesting and relevant. Erik Quaeghebeur was supported by a Fellowship of the Belgian American Educational Foundation and wishes to thank Carnegie Mellon University’s Department of Philosophy for its hospitality.

Keywords

Exchangeability
Extending an exchangeable sequence
Natural extension
Representation
Sets of desirable gambles
Updating

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10.1016/j.ijar.2010.12.002

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AB - Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We give a general discussion of such models and their rationality criteria. We study exchangeability assessments for them, and prove counterparts of de Finetti's Finite and Infinite Representation Theorems. We show that the finite representation in terms of count vectors has a very nice geometrical interpretation, and that the representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bare the relationships between the representations of updated exchangeable models, and discuss conservative inference (natural extension) under exchangeability and the extension of exchangeable sequences.

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KW - Extending an exchangeable sequence

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KW - Representation

KW - Sets of desirable gambles

KW - Updating

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ER -

Exchangeability and sets of desirable gambles

Abstract

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Keywords

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