ROSS D. SHACHTER wrote:
>That way I am open to learning something from what you have to say.
>Could you please explain to me how if P{R|Q} = 0.9 and P{R|�Q} = 0.8 you
>can have a Bel(R) less than 0.8 or a Pl(R) bigger than 0.9?
Read the paper I published in IJAR in the second issue on belief fct (the
1st had the paper by Judea, and the second a long reply where I show that
the critics were inadequate.
The whole point is to understand what is belief.
To Bruce: why do you still stick to bel being a lower bound. Bound of what?
A proba, OK but then move toward interval value probability, in which case
belief fct, TBM and Dempster model are unrelated.
Juda Pearl Comments.
The latest discussion of belief functions (BF) has reminded me of
the difficult time I had in 1987-88, when I struggled to
understand the Dempster-Shafer (D-S) theory and to relate it
to Bayesian analysis. Fortunately for me, that struggle
has ended in a simple interpretation of belief functions
that settled all my difficulties,
put my mind at ease, and clearly delineated the type of
applications where BF would be useful.
This interpretation, which identifies BF's with
"PROBABILITY OF NECESSITY" (or probability of provability),
has helped me turn endless philosophical arguments into fruitful
discussions with the proponents of D-S theory.
I think it can also add clarity to the
latest discussion that bloomed from Smets' innocent remark.
Judea is correct when he states that belief fct is exactly what you get
when you start applying proba theory to propositions in modal logic instead
of propositions in classical logic. Just call bel(A) as P(boxA) and you
will find out that bel is a belief fct. Where Judea is optimist is when
conditioning on some B is introduced. Should we update the accessibility
relation (in which case we end up with unnormalized Dempster rule of
conditioning) or should we put to 0 the probability of the worlds where B
is not provable (in which case one ends up with the so called geometric
rule of conditioning).
In the TBM we don't even consider proba, but I agree that seeing bel(A) as
proba of knowing A is true (as defined in Ruspini in +/-86) or proba of
proving A is true (as Judea proposed) or proba of necessity of A is true
(what several people in the fuzzy community have suggested) is often
helpful.
Again Judea said:
I disagree.
The main difference
between the Bayesian and the Belief Function approach is
that the former aims at the probability of facts
and the latter changes the question around, and aims at the
probability of having a proof for the facts (given no-contradiction).
Well THIS is Judea's approach, NOT the TBM. The TBM intends to quantify the
strength of my opinions, not a matter of probability. In fact Judea's
approach is close to Rolf Haenni and Cie approach, but I don't assimilate
this approach to the TBM. In Dempster, Kohlas, Haenni etc� you always start
with a proba fct on some space� and end up with a bel fct on another space.
My question: why to start with a proba on the first space, and not a bel
fct.
THE question is to decide if proba is really the correct model to represent
beliefs. Yes claim Bayesians, and their argument is usually not based on
Cox's axioms but on betting behavior, Dutch Books and money pump arguments.
These arguments all justify the need of a proba measure for decision making
(and this is my pignistic proba) but it has never justify that beliefs must
also be represented by a proba fct except if one claims that entertaining
beliefs without taking decision does not exist). So the problem is to
justify proba at my so called credal level. Cox's axioms could be used
there, but why do we have to acknowledge the axiom for the negation? So I
try to develop a set of constraints that belief measure should satisfy,�
and derive bel fct in a 94' IJACI paper and 97'AI paper (good reading).
Rich proposes:
Sandy Zabell discusses this problem in detail and has some
recent concrete results. I will give the references if anyone is interested.
Yes I am. Thanks.
All the best all f you
Philippe
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Philippe Smets
email: [EMAIL PROTECTED]
IRIDIA-CP 194/6
Universite Libre de Bruxelles
50 av. Roosevelt,
1050 Bruxelles, Belgium.
tel 32 2 650 27 29 secretary,
32 2 344 82 96 private (where I am usually)
fax: 32 2 650 27 15
GSM: 32 495 50 10 72
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