Information Fusion in Signal and Image Processing: Major Probabilistic and Non-Probabilistic Numerical Approaches

Appendix B

Axiomatic Inference of the Dempster-Shafer Combination Rule

The Dempster-Shafer belief theory has often been criticized for imposing an ad hoc combination rule (Dempster's orthogonal rule [SHA 76], equation [7.26]), with no theoretical justification for it, although it does have a perfectly satisfactory intuitive interpretation, which is in agreement with the concept of the conjunction of focal elements.

This appendix completes the theory presented in Chapter 7 by giving a theoretical justification of the conjunctive combination rule (non-normalized, as it is suggested in [SME 90]).

We will also explain how it is related to the Cox approach to probabilities, which is presented in Appendix A. In particular, we will underline the differences between the axioms, which explain the difference between the models and the combination modes in the two theories.

Several recent works have attempted to justify this rule, for example, those of Dubois and Prade [DUB 86], which mathematically justify the use of the product to combine masses based on the concept of the separability of sources, or those of Smets based on the transferable belief model [SME 90].

We should also mention the works of Gacôgne, which have led to a justification, in the specific cases where the frame of discernment is reduced to two elements, based on the concept of accentuation [GAC 93]¹.

The works of Smets offer the most general justification, as far as we know, and his arguments will be described here. Furthermore, ...