Dempster--Shafer Theory and Statistical Inference with Weak Beliefs

TL;DR

This paper reviews and extends the weak belief approach, an enhancement of Dempster--Shafer theory, demonstrating its effectiveness in high-dimensional hypothesis testing with favorable frequency properties and competitive performance.

Contribution

It introduces new applications of the weak belief method in high-dimensional testing and shows how it combines DS reasoning with classical frequency properties.

Findings

WB procedures perform well compared to classical methods

Simulations confirm the effectiveness of WB in high-dimensional tests

WB approach successfully incorporates frequency properties into DS framework

Abstract

The Dempster--Shafer (DS) theory is a powerful tool for probabilistic reasoning based on a formal calculus for combining evidence. DS theory has been widely used in computer science and engineering applications, but has yet to reach the statistical mainstream, perhaps because the DS belief functions do not satisfy long-run frequency properties. Recently, two of the authors proposed an extension of DS, called the weak belief (WB) approach, that can incorporate desirable frequency properties into the DS framework by systematically enlarging the focal elements. The present paper reviews and extends this WB approach. We present a general description of WB in the context of inferential models, its interplay with the DS calculus, and the maximal belief solution. New applications of the WB method in two high-dimensional hypothesis testing problems are given. Simulations show that the WB procedures, suitably calibrated, perform well compared to popular classical methods. Most importantly, the WB approach combines the probabilistic reasoning of DS with the desirable frequency properties of classical statistics.