Hello All :-
In reply to Kevin's inquiry
> Has anyone done the analog of Cox's Theorem for belief function theories
> or possibility theory? That is, has anyone postulated a different set of
> qualitative requirements and then shown that these requirements lead
> inevitably to a system isomorphic to one of these non-Bayesian theories?
and with regard to possibility theory:
Linear possibility (all atomic values distinct) is a _solution_ to Cox's
1946 equations. This was published in Computational Intelligence last year
(vol. 17, 178-192, Feb 2001). LP is also shown to be syntactically related to
the ordinal relationships in an "atomic bound" probability (typical example:
1/31, 2/31, 4/31, 8/31, 16/31).
Cox's development of Keynesian probabilistic ideas (especially variable
and set-valued belief representation) cover the other possibilistic cases.
There are any number of axiomatic derivations of possibility which do not
rely on probability. Philippe posted something about this on the list a few
months ago, for example.
I also like the one in which v( a or b ) = max [ v(a), v(b) ] is the
unique ordering rule which represents both "belief" in the Lukasiewicz-Sugeno
sense:
if x implies y, then v(y) >= v(x)
and is also consistent with elementary and weak notions of preference
orderings (and "support" in the conditional probability of evidence sense):
if x implies y, then v(y) > v(x) only if v(y&-x) > v(x)
I have used this in talks, and it is in a paper forthcoming at FLAIRS.
There is also a quasi-additive max calculus (a partial order, and L-S
only) which is somewhat similar to possibility, and has applications in some
statistical work as well as default reasoning. An axiomatic development of
that one is in IEEE Trans. SMC from May 1996 (349-360).
Hope that this is helpful.
Paul Snow
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