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ISIPTA'07 -
FIFTH INTERNATIONAL SYMPOSIUM ON

IMPRECISE PROBABILITY: THEORIES AND APPLICATIONS

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Charles University, Faculty of Mathematicsand Physics

Prague, Czech Republic

16-19 July 2007

## ELECTRONIC PROCEEDINGS

## Gert De Cooman, Filip Hermans

# On coherent immediate prediction: Connecting two theories of imprecise probability

### Abstract

We give an overview of two approaches to probability theory where lower and upper probabilities, rather than probabilities, are used: Walley's behavioural theory of imprecise probabilities, and Shafer and Vovk's game-theoretic account of probability. We show that the two theories are more closely related than would be suspected at first sight, and we establish a correspondence between them that (i) has an interesting interpretation, and (ii) allows us to freely import results from one theory into the other. Our approach leads to an account of immediate prediction in the framework of Walley's theory, and we prove an interesting and quite general version of the weak law of large numbers.

** Keywords. ** Game-theoretic probability, imprecise probabilities, coherence, conglomerability, event tree, lower prevision, immediate prediction, Prequential Principle, law of large numbers, Hoeffding's inequality.

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** Authors addresses: **

Gert De Cooman

Technologiepark - Zwijnaarde 914

9052 Zwijnaarde

Filip Hermans

Technologiepark-Zwijnaarde 914

9052 ZWIJNAARDE (GENT)

** E-mail addresses: **

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