Errors bounds for finite approximations of coherent lower previsions on finite probability spaces

TL;DR

This paper introduces a practical method to estimate the maximum error when approximating coherent lower probabilities on finite spaces, addressing a gap in error quantification for probabilistic models.

Contribution

It provides a new, applicable approach to calculate upper bounds on approximation errors for coherent lower previsions, including an algorithm with complexity analysis.

Findings

The method effectively bounds the maximum approximation error.

The algorithm's computational complexity is estimated and analyzed.

The approach enhances the reliability of finite approximations in probabilistic modeling.

Abstract

Coherent lower previsions are general probabilistic models allowing incompletely specified probability distributions. However, for complete description of a coherent lower prevision -- even on finite underlying sample spaces -- an infinite number of assessments is needed in general. Therefore, they are often only described approximately by some less general models, such as coherent lower probabilities or in terms of some other finite set of constraints. The magnitude of error induced by the approximations has often been neglected in the literature, despite the fact that it can be significant, with substantial impact on consequent decisions. An apparent reason is that no widely used general method for estimating the error seems to be available at the moment. This paper provides a practically applicable method that allows calculating an upper bound for the maximal error induced by approximating a coherent lower probability with its values on a finite set of gambles. An algorithm is also provided with an estimation of its computational complexity.