Imprecise probability trees: Bridging two theories of imprecise probability

TL;DR

This paper explores the relationship between two theories of imprecise probability, establishing a correspondence that enables results transfer and simplifies computations in probability trees.

Contribution

It bridges Walley's and Shafer-Vovk's theories, allowing for a unified framework and improved computational methods for imprecise probability models.

Findings

Established a correspondence between the two theories of imprecise probability.

Developed a probability tree framework within Walley's theory.

Proved a general version of the weak law of large numbers.

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 probability trees and random processes in the framework of Walley's theory. We indicate how our results can be used to reduce the computational complexity of dealing with imprecision in probability trees, and we prove an interesting and quite general version of the weak law of large numbers.