Pr(Q) is some unknown number p between 0 and 1
Pr(A1)=0.9, independent of Q and A2
Pr(A2)=0.8, independent of Q and A1
Pr(R|Q,A1)=1.0
Pr(R|~Q,A2)=1.0
Pr(R|Q)=0.9
Pr(R|~Q)=0.8
If all these are accurate, then here is the complete table for Pr(R|Q,A1,A2):
Pr(R|Q,A1,A2)=1
Pr(R|Q,A1,~A2)=1
Pr(R|Q,~A1,A2)=0*
Pr(R|Q,~A1,A2)=0*
Pr(R|~Q,A1,A2)=1
Pr(R|~Q,A1,~A2)=0**
Pr(R|~Q,~A1,A2)=1
Pr(R|~Q,~A1,~A2)=0**
*These have to be true because Pr(R|Q)=Pr(R|Q,A1)=0.9
** These have to be true because Pr(R|~Q)=Pr(R|~Q,A2)=0.8
Now it is true that P(R|A1,A2)=1 and Pr(R|~A1,~A2)=0.
But just because we don't know the correct value, that doesn't mean we can't
say anything about Pr(R|A1,~A2) and Pr(R|~A!,A2). In particular, even though
we don't know the value of p, we do know that
Pr(R|A1,~A2) = .p
Pr(R|~A1,A2) = 1-p
Therefore, we have the following complete expression for Pr(R):
Pr(R) = 0.72x1 + 0.18p + 0.08(1-p) + 0.02x0 = 0.8 + 0.1p
Recalling that 0 <= p <= 1, it follows that 0.8 <= Pr(R) <= 0.9.
Moral: just because you can't say anything about X and you can't say anything
about Y, that doesn't mean you can't say anything about the pair (X,Y).
Note: In the past, I have seen examples like this used to refute
non-probabilistic belief calculi as incoherent. I think it more
appropriate to
say that the particular methods that have been proposed may be incoherent to
the degree that they ignore (and often obscure) constraints imposed by logic,
but that when these constraints are suitably represented and adhered to,
alternate belief calculi may be valid extensions that include probability
theory as a special limiting case, just as probability includes binary
logic as
a special limiting case.
Jonathan Weiss
At 6/8/99 10:33 AM, Rolf Haenni wrote:
>Dear UAI community, dear Ross D. Shachter,
>
>
>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
>
>
>
>************************************************************************
>* *
>* 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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<http://www2-iiuf.unifr.ch/tcs/rolf.haenni>http://www2-iiuf.unifr.ch/tcs/rol
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