Independence, Conditionality and Structure of Dempster-Shafer Belief Functions

TL;DR

This paper reviews various methods of structuring Dempster-Shafer belief functions, analyzing their ability to represent independence and dependence akin to Bayesian networks, and introduces a unified approach that captures more independences.

Contribution

It compares existing frameworks for belief function factorization and proposes a new approach that combines their strengths to better represent variable independence.

Findings

Existing models fail to fully capture Bayesian independence.

Shenoy and Shafer's hypergraphs can represent Bayesian factorizations.

The proposed approach captures more independences than previous models.

Abstract

Several approaches of structuring (factorization, decomposition) of Dempster-Shafer joint belief functions from literature are reviewed with special emphasis on their capability to capture independence from the point of view of the claim that belief functions generalize bayes notion of probability. It is demonstrated that Zhu and Lee's {Zhu:93} logical networks and Smets' {Smets:93} directed acyclic graphs are unable to capture statistical dependence/independence of bayesian networks {Pearl:88}. On the other hand, though Shenoy and Shafer's hypergraphs can explicitly represent bayesian network factorization of bayesian belief functions, they disclaim any need for representation of independence of variables in belief functions. Cano et al. {Cano:93} reject the hypergraph representation of Shenoy and Shafer just on grounds of missing representation of variable independence, but in their frameworks some belief functions factorizable in Shenoy/Shafer framework cannot be factored. The approach in {Klopotek:93f} on the other hand combines the merits of both Cano et al. and of Shenoy/Shafer approach in that for Shenoy/Shafer approach no simpler factorization than that in {Klopotek:93f} approach exists and on the other hand all independences among variables captured in Cano et al. framework and many more are captured in {Klopotek:93f} approach.%