Interpreting evidential distances by connecting them to partial orders: Application to belief function approximation

• Distances and partial orders in the theory of belief functions are formally related through a desirable property.
• The compatibility of several LkLk norm based distances with some informational partial orders is proved.
• These results are exploited to approximate initial belief functions by simpler ones through the use of convex optimization.

Distances between mass functions are instrumental tools in evidence theory, yet it is not always clear in which situation a particular distance should be used. Indeed, while the mathematical properties of distances have been well studied, how to interpret them is still a largely open issue. As a step towards answering this question, we propose to interpret distances by looking at their compatibility with partial orders. We formalize this compatibility through some mathematical properties thereby allowing to combine the advantages of both partial orders (clear semantics) and distances (richer structure and access to numerical tools). We explore in particular the case of informational partial orders, and how distances compatible with such orders can be used to approximate initial belief functions by simpler ones through the use of convex optimization. We finish by discussing some perspectives of the current work.