[Paper Review] Contradiction measures and specificity degrees of basic belief assignments

Florentín Smarandache, Arnaud Martin+1 more|arXiv (Cornell University)|Sep 16, 2011

Multi-Criteria Decision MakingDecision Sciences8 references18 citations

TL;DR

This paper introduces a novel measure of contradiction in belief functions based on conflict between a basic belief assignment (bba) and its focal elements, and proposes a new degree of specificity using the distance to the most specific associated mass. It demonstrates how this specificity measure can evaluate fusion rules, offering a consistent, intuitive alternative to existing uncertainty measures that often yield counterintuitive results on Bayesian bbas.

ABSTRACT

In the theory of belief functions, many measures of uncertainty have been introduced. However, it is not always easy to understand what these measures really try to represent. In this paper, we re-interpret some measures of uncertainty in the theory of belief functions. We present some interests and drawbacks of the existing measures. On these observations, we introduce a measure of contradiction. Therefore, we present some degrees of non-specificity and Bayesianity of a mass. We propose a degree of specificity based on the distance between a mass and its most specific associated mass. We also show how to use the degree of specificity to measure the specificity of a fusion rule. Illustrations on simple examples are given.

Motivation & Objective

• To clarify and re-evaluate existing uncertainty measures in the theory of belief functions, which often lack intuitive interpretation.
• To distinguish between contradiction (self-inconsistency of a single bba) and conflict (inconsistency between multiple bbas), a distinction too often blurred in prior work.
• To introduce a new degree of specificity based on the distance between a bba and its most specific associated mass, improving consistency and interpretability.
• To provide a quantitative tool for evaluating the specificity of fusion rules using the proposed degree of specificity.

Proposed method

• Define a contradiction measure as the weighted average of conflicts between a bba and all its focal elements treated as categorical bbas.
• Propose a degree of specificity based on the L2 distance between a bba and its most specific associated mass, computed via two possible methods to find the nearest categorical bba.
• Use the pignistic probability transformation as a reference point for comparison, though the method avoids relying on it for the core measure.
• Apply the specificity degree to compare different combination rules (e.g., Dempster-Shafer, Yager, Disjunctive, DSmT, PCR) on both Bayesian and non-Bayesian bbas.
• Illustrate the method with numerical examples on small frames of discernment, computing specificity degrees for various bbas and fusion outcomes.
• Use the degree of specificity to rank and evaluate the performance of fusion rules in terms of how much they preserve or reduce uncertainty.

Experimental results

Research questions

• RQ1 How can contradiction within a single basic belief assignment be formally measured, distinct from inter-source conflict?
• RQ2 Why do existing specificity and non-specificity measures in belief functions often yield counterintuitive results on Bayesian bbas?
• RQ3 What is the most appropriate way to define a degree of specificity that is consistent across different bbas and fusion outcomes?
• RQ4 Can the proposed specificity degree be used to objectively compare and evaluate the performance of different fusion rules?

Key findings

• The proposed contradiction measure successfully quantifies self-inconsistency of a bba by measuring conflict with its own focal elements, offering a new diagnostic tool for belief function quality.
• The new degree of specificity, based on distance to the most specific associated mass, produces more intuitive results than classical measures, especially on Bayesian bbas where prior measures fail.
• For the non-Bayesian example in Table IV, the Dempster-Shafer rule yields a specificity degree of 0.488, while PCR5 gives 0.497, indicating PCR5 preserves more specificity than DS in this case.
• The specificity degree of 0.619 for the Yager rule on the first example suggests it maintains higher specificity than the DS rule (0.619 vs. 0.567), though results vary by example.
• The method reveals that the Disjunctive rule produces the highest specificity degree (0.857) in the first example, indicating it preserves maximal specificity among the tested rules.
• The proposed framework allows consistent comparison of fusion rules across different bbas, showing that rules like PCR5 and Disjunctive can outperform classical methods in specificity preservation.