Probability boxes on totally preordered spaces for multivariate modelling

TL;DR

This paper extends the theory of probability boxes (p-boxes) to multivariate cases on totally preordered spaces, providing new tools and algorithms for inference, with applications to engineering problems.

Contribution

It formalizes multivariate p-boxes using Walley's imprecise probability theory and develops algorithms for inference under independence and unknown dependence.

Findings

Extended p-box theory to arbitrary totally preordered spaces.

Developed algorithms for inference with multivariate p-boxes.

Demonstrated practical applications on engineering problems.

Abstract

A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley's behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fr\'echet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems---a damped oscillator, and a river dike---demonstrate the practical feasibility of our results.