thanks to Judea's help clarifying the situation by identifying the notion
of BELIEF as the "PROBABILITY OF NECESSITY" (or probability of
provability). I agree with this point of view. In fact, in probabilistic
argumentation systems, instead of BELIEF we prefer to say DEGREE OF
SUPPORT, which is defined as the probability of the supporting arguments (=
possible proofs). More precisely, it's a conditional probabilitiy given no
contradiction.
Unfortunately, I don't have the time to reply to every individual point
discussed in the emails I received today. However, I see that proponents of
the Baysian approach have big difficulties to accept a value Bel(R)=0.72
lower than 0.8 and a value Pl(R)=0.98 higher than 0.9, as Kathryn B. Laskey
said:
>One can "explain" the phenomenon by saying that there is only a 0.72 chance
>that "the evidence would prove R," but I was never able to come up with a
>way to argue this convincingly to a subject matter expert. I guess that's
>because I can't argue it convincingly to myself. I can follow the
>mathematics, but I don't have a handle on what it means.
Let me try to clarify this further. As I already said, the crucial point is
the total ignorance about Q. Total ignorance means YOU DON'T KNOW ANYTHING
ABOUT Q, i.e. first of all, you don't know a prior probability (see my
example about the existence of god), but secondly, it also means that you
don't even know whether such an (independent) prior probability exists.
Note that the truth of Q could possibly depend on R, or on A1 or on A2. YOU
DON'T KNOW IT.
For example, if Q depends on A1 and A2 in the following way:
1) (A1 and ¬A2) --> ¬Q,
2) (¬A1 and A2) --> Q,
then we get Bel(R)=Pl(R)=P(R)=0.72 and Bel(¬R)=Pl(¬R)=P(¬R)=0.28. In
contrast, if Q depends on A1 and A2 by
1) (A1 and ¬A2) --> Q,
2) (¬A1 and A2) --> ¬Q,
then Bel(R)=Pl(R)=P(R)=0.98 and Bel(¬R)=Pl(¬R)=P(¬R)=0.02. This shows how
the probabilities for the cases 2) and 3) can simultaneously jump either to
R or to ¬R (see my last email):
1) A1 and A2 --> R is automatically true (0.72)
2) A1 and ¬A2 --> nothing can be said about R (0.18)
3) ¬A1 and A2 --> nothing can be said about R (0.08)
4) ¬A1 and ¬A2 --> ¬R is automatically true (0.02)
To summarize, if nothing is known about Q (not even whether an independent
prior probability exists), then it makes perfectly sense to say that the
Belief (or the probability of the provability) is 0.72. The intuition that
the value must be between 0.8 and 0.9 comes from the assumption that a
prior probabilty exists.
This is finally the main point producing all the confusion. It's clear,
that for a proponent of the Bayesian approach is perhaps difficult to give
up the assumption that prior probabilities exist. However, I think it's
necessary in order to capture the nature of total ignorance properly.
The message of K.S.Van Horn underlines all this:
KEVIN S. VAN HORN wrote:
>...regardless of the value of P(Q), we know from 0 <= P(Q) <= 1 that 0.8 <=
>P(R) <= 0.9.
==> as I said, it may be difficult to give it up!!! :-)
KEVIN S. VAN HORN wrote:
>Again, Haenni's theory is losing information by giving unnecessarily
>loose bounds.
==> or should we say, YOU are ADDING information??? :-)
To conclude, I think it should be clear now that the main difference
between the Bayesian and the Belief Function approach is just given by the
way in which total ignorance is handled. For me, the "existence of
God"-example is a strong indication that total ignorance is handled more
accurately by belief functions (and also by probabilistic argmentation
systems), that's all.
Enjoy your day,
Rolf Haenni
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* *
* Dr. Rolf Haenni __/ __/ __/ __/ _______/ *
* Institute of Informatics (IIUF) __/ __/ __/ __/ __/ *
* University of Fribourg, Switzerland __/ __/ __/ __/ _____/ *
* Phone: ++41 26 300 83 31 __/ __/ __/ __/ __/ *
* Email: rolf.haenni@unifr.ch __/ __/ ______/ __/ *
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