Likelihood-based belief function: Justification and some extensions to low-quality data

Thierry Denœux

doi:10.1016/j.ijar.2013.06.007

Abstract

Given a parametric statistical model, evidential methods of statistical inference aim at constructing a belief function on the parameter space from observations. The two main approaches are Dempster's method, which regards the observed variable as a function of the parameter and an auxiliary variable with known probability distribution, and the likelihood-based approach, which considers the relative likelihood as the contour function of a consonant belief function. In this paper, we revisit the latter approach and prove that it can be derived from three basic principles: the likelihood principle, compatibility with Bayes' rule and the minimal commitment principle. We then show how this method can be extended to handle low-quality data. Two cases are considered: observations that are only partially relevant to the population of interest, and data acquired through an imperfect observation process.

Highlights

Belief functions were initially introduced by Dempster as part of a new approach to statistical inference, in which lower and upper posterior probabilities can be constructed without having to postulate the existence of prior probabilities [10, 11, 12]
In support of the method outlined above for constructing belief functions from data, we can remark that viewing the relative likelihood function as the contour function of a consonant belief function or, equivalently, as a possibility distribution [48, 21] is, to a large extent, consistent with statistical practice
Likelihood intervals [29, 42] are focal intervals of the relative likelihood viewed as a possibility distribution

Summary

Introduction

Belief functions were initially introduced by Dempster as part of a new approach to statistical inference, in which lower and upper posterior probabilities can be constructed without having to postulate the existence of prior probabilities [10, 11, 12]. Dempster’s initial ideas were later formalized by Shafer [37] and transformed into a general system for reasoning under uncertainty, usually referred to as Dempster-Shafer theory. New variants of Dempster’s method of inference leading to belief functions having some well-defined long-run frequency properties have been proposed, such as the Weak Belief approach [32], the Elastic Belief method [30] and the Inferential Models [31]. The key idea underlying Dempster’s method of inference and its variants is to represent such a sampling model by an equation

Objectives

Discussion

Conclusion