>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.
This example illustrates one of my fundamental difficulties with belief
functions: the phenomenon my colleague Anne Martin picturesquely labeled
"leaky belief." Here we have a situation in which the probability of R is
in either case (given Q or given not-Q) greater than 0.8 and less than 0.9.
Nevertheless, the "bounds" given by Bel(R) and Pl(R) are 0.72 to 0.98.
(Anne used to wonder how the belief "leaked out" and where it went.) I
understand that these bounds should not be interpreted as upper and lower
probabilities. Nevertheless, the "leaky belief" phenomenon seems odd to
me. No matter what prior probability we put on Q, the marginal probability
of R in any probability model would lie between 0.8 and 0.9.
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.
Moreover, by varying the dependence structure of A1 and A2, I can construct
belief functions with beliefs Bel(R) varying from 0.7 to 0.8, and with
corresponding plausibilities Pl(R) varying from 1.0 to 0.9. One might
argue that the values Bel(R) =0.8 and Pl(R) = 0.9 are the "right" ones,
given that they correspond to the bounds of the probability model. (These
values occur when A1 and A2 are maximally positively correlated given the
marginal distributions.) However, I could never generalize this intuition
to more complex cases. I also never managed to come up with a semantic
interpretation of A1 and A2 that was convincing to me. In the absence of a
semantic interpretation, it's hard to see what the justification is for any
particular assumption about their dependence structure.
Perhaps it was just my lack of creativity that prevented me from ever
coming up with a satisfying semantics for belief functions, that would lead
to resolutions of modeling difficulties like this one. If so, I'd like to
be educated. But in the meantime, I'll stick to probability. At least
when I get strange answers, I feel confident the fault lies with my model
and not the mapping of my mathematical formalism to a useful semantic
notion of reasonable belief.
Kathy Laskey