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?
Yes, let me try to explain it. P(R|Q)=0.9 tells us that given Q, R is also
true in 90% of the cases. In other words, R is only true given Q, if an
additional condition A1 is true with P(A1)=0.9. Therefore, A1 represents
somehow the uncertainty of the implication Q-->R. Similarly, A2 with
P(A2)=0.8 represents the uncertainty of the implication ¬Q-->R. Assuming
that A1 and A2 are true at the same time implies automatically that R is
true without knowing anything about Q. The conjunction <A1 and A2> can
therefore be seen as a sufficient proof (or a "supporting argument", as we
prefer to say in the framework of probabilistic argumentation systems) for
the truth of R. The probability of this proof,
P(A1 and A2) = 0.8 * 0.9 = 0.72,
corresponds then to Bel(R)=0.72. Similarly, assuming that ¬A1 and ¬A2 are
true at the same time implies automatically ¬R without knowing anything
about Q. So, the conjunction <¬A1 and ¬A2> can be seen as proof of ¬R, and
therefore
Pl(R) = 1 - Bel(¬R) = 1 - 0.2*0.1 = 0.98.
To make the situation more clear, note that in the case of total ignorance
about Q, there are 4 situations:
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)
The difference between 0.72 and 0.98 comes from the cases 2) and 3), where
nothing can be concluded about R. It may look strange to get a value <0.8
and a value >0.9, but I think it reflects somehow the nature of the total
ignorance about Q.
I don't know if this is very convincing, but it's my way of understanding
it from the point of view of probabilistic argumentation systems.
Thanks to everybody participating or reading this interesting discussion.
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 __/ __/ __/ __/ __/ *
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