Philippe QUINIO
Mathematical Connections between the probability, Fuzzy set, Possibility and Dempster-Shafer theories
Abstract:The authors of "new theories" for the representation of uncertainty, imprecision or fuzziness in AI, including the Upper/lower probability framework, Dempster-Shafer (DS) theory of Evidence (Belief/Plausibility functions), Possibility theory (Possibility/Necessity measures), Fuzzy set and Random Closed Set (RACS) theories spent much of their energy trying to isolate them and prevent external criticism or comparison with other scientific theories.
We believe however that the time has come for a systematic in-depth theoretical comparison. The mathematical links between the Random Closed Set formalism and topological versions of the above mentioned theories are investigated, the basis for comparison being purely axiomatic. The RACS theory emerges as a sufficient conceptual and mathematical framework for the representation of uncertainty, imprecision and fuzziness. The underlying topological setting makes it sufficiently general so as to encompass all AI problems, but not so general so as to include useless, experimentally inaccessible mathematical abstractions.
Then this paper provides a discussion about the necessity and the usefulness of theoretical comparisons in the context of Artificial Intelligence, with a particular emphasis on the axiomatics/interpretation dilemma.
Since the RACS theory is merely an application of general probability measure theory to a topologically meaningful subpart of the power set of a Universe U equipped with a topology derived from that of U itself, the tone of this paper might appear somewhat retrograde. We insist, however, on the practical usefulness of the above theoretical comparisons, that go far beyond mere scientific conscientiousness. And we show how such comparisons can be exploited in order to compensate for theoretical weaknesses in the above formalisms and yield useful hybrid techniques. An important result characterizes the Mean probabilistic operation as the only order-independent, piecewise and point-compatible combination for constructing general Belief, Plausibility or Fuzzy set membership functions from subsets of a Universe. This reduces the possible choices of a construction scheme in systems where the sources of information are not human, or at least not only human.