Ploxoma: Testbed for Uncertain Inference

This paper compares two formalisms for uncertain inference, Kyburg's Combinatorial Semantics and Dempster-Shafer belief function theory, on the basis of an example from the domain of medical diagnosis. I review Shafer's example about the imaginary disease ploxoma and show how it would be represented in Combinatorial Semantics. I conclude that belief function theory has a qualitative advantage because it ooers greater exibility of expression, and provides results about more speciic classes of patients. Nevertheless, a quantitative comparison reveals that the inferences sanctioned by Combinatorial Semantics are more reliable than those of belief function theory. 5.1 Introduction This paper compares Kyburg's Combinatorial Semantics (CS) and Dempster-Shafer belief function theory (BFT), two formalisms for representing uncertain inference from statistical data. I reconsider a medical diagnosis example introduced by Shafer Shafer82] concerning the imaginary disease ploxoma. In Section 2, I cite Shafer's example and sketch the steps of its treatment using BFT. Section 3 presents a pr ecis of Combinatorial Semantics 2 , and Section 4 shows how the ploxoma example can be handled in the CS framework. Section 5 draws conclusions from this comparison. One of the operations applied in the BFT analysis cannot be translated into CS, and CS groups together two classes of patients that BFT can distinguish. I compute new belief functions after eliminating from the example those elements that are problematic for CS. While BFT appears on qualitative grounds to be more exible and informative, a quantitative comparison shows that the conclusions sanctioned by CS are more reliable. BFT's results may not conform to frequencies observed in the real world; those of CS do. Many authors, including Hunter87, Lemmer86, Pearl90], have observed that belief functions lack a simple frequency interpretation. The contribution of this paper is a comparison between BFT and CS, discussed here for the rst time. In the tradition of Smets Smets91, Smets94], I center my investigation on a concrete instance chosen not as a counterexample, nor as a philosophical conundrum like the three prisoner problem Diaconis78, DiaconisZabell86], but as a basis for contrasting alternative analyses. A good example is a valuable tool to isolate the points of diierence among theories, and to identify their strengths and weaknesses. Shafer's ploxoma story suits this purpose well because it is 1996 Springer-Verlag. 2 Combinatorial Semantics Kyburg95] provides the semantic underpinnings for Evidential Probability Kyburg83].