The Goodman-Nguyen Relation within Imprecise Probability Theory

TL;DR

This paper explores the Goodman-Nguyen relation's role in imprecise probability theory, extending its application from conditional events to gambles, and demonstrating its utility in ordering and inferential inequalities.

Contribution

It introduces a generalization of the Goodman-Nguyen relation to conditional gambles within imprecise probability, expanding its theoretical and practical applications.

Findings

Induces an agreeing ordering on imprecise previsions.

Enables determination of natural and upper extensions in inference.

Useful for deriving inferential inequalities with conditional gambles.

Abstract

The Goodman-Nguyen relation is a partial order generalising the implication (inclusion) relation to conditional events. As such, with precise probabilities it both induces an agreeing probability ordering and is a key tool in a certain common extension problem. Most previous work involving this relation is concerned with either conditional event algebras or precise probabilities. We investigate here its role within imprecise probability theory, first in the framework of conditional events and then proposing a generalisation of the Goodman-Nguyen relation to conditional gambles. It turns out that this relation induces an agreeing ordering on coherent or C-convex conditional imprecise previsions. In a standard inferential problem with conditional events, it lets us determine the natural extension, as well as an upper extension. With conditional gambles, it is useful in deriving a number of inferential inequalities.