A betting interpretation for probabilities and Dempster-Shafer degrees of belief

TL;DR

This paper explores two betting-based interpretations of belief degrees, showing how they justify probability updating and Dempster-Shafer combination, with implications for understanding uncertainty measures.

Contribution

It introduces a betting interpretation framework that clarifies the foundations of probabilities and Dempster-Shafer belief functions, linking them to practical betting strategies.

Findings

Betting interpretation justifies probability updating by conditioning.

Only the second betting interpretation applies to Dempster-Shafer degrees.

Dempster's rule of combination is justified through the second interpretation.

Abstract

There are at least two ways to interpret numerical degrees of belief in terms of betting: (1) you can offer to bet at the odds defined by the degrees of belief, or (2) you can judge that a strategy for taking advantage of such betting offers will not multiply the capital it risks by a large factor. Both interpretations can be applied to ordinary additive probabilities and used to justify updating by conditioning. Only the second can be applied to Dempster-Shafer degrees of belief and used to justify Dempster's rule of combination.