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?
I've always had a somewhat different understanding of this problem.
The first important detail is that although in probability specifing
P(R|Q) and P(R|~Q) along with P(Q) is a complete specification for the
joint distribution P(R,Q), Bel(R|Q), Bel(R|~Q) and Bel(Q) is not
sufficient to produce a unique joint belief function over R & Q. In
order to get a unique solution, you would need to add additional
constraints.
My choice of constaint would be adding Bel(R|Q or ~Q) = .8 and Pl(R|Q
or ~Q) = .9. You will then get something which is more consistant
with what you expect.
Actually, if I were seriously looking at this problem. I might encode
my knowledge as:
Bel(R|Q) = Pl(R|Q) = .9
Bel(R|Q or ~Q) = .8
Pl(R|Q or ~Q) = .9
I think these better express what is going on and produce a joint
believe with the same values.
Another approach is to assume that Bel(R|Q) and Bel(R|~Q) are
independent. I think Rolf Haenni did a pretty good job of describing
that approach. It really doesn't make much sense unless you introduce
the additional idea of propositions A1 and A2 which prove the logical
propositions. You then further assert that A1 and A2 are independent
(in the formal statistical sense). This is what the assertion that
Bel(R|Q) and Bel(R|~Q) boils down to. Walley is critical of it as
being very abstract and hence confusing, and I think I agree.
I've seen the burried independence assumption in Smetts' rule of
conditional embedding catch many people off guard. It seems to work
as a construction method if you follow the belief function methodology
and specify conditional belief for every set of configurations of the
conditioning variables. However, if you try and only condition on
single configuration sets for the conditioning variables you can get
surprizing results.
LOCAL INDEPENDENCE
I think we all agree that the indepedence assumption for Bel(R|Q) and
Bel(R|~Q) has some surprizing implications. However, I've run across
it in a different form in the so-called "local independence
assumption."
This one states that if
P(R|Q) = p1
P(R|~Q) = p0
where p1 and p0 are unknown (and hence random variables if we are
Bayesians) then p1 and p0 are independent. This bothers me, in large
part because I know what kind of trouble the similar independence
assumption gives in the belief function problem.
In particular, under local independence
Pr { P(R|Q) < P(R|~Q) } > 0
This may our may not be what I intended.
Often I think a better parameterization is:
P(R|~Q) = p0
P(R|Q) = p0 + (1-p0)q1 = p1
Where p0 and q1 are independent random variables over [0,1]. This has
the effect that
Pr{ P(R|Q) < P(R|~Q) } = 0.
There is an interesting class of bivariate beta distributions for p0
and p1 which can be described with three parameters. Unfortunately,
it is not fully conjugate with P(R,Q) and you need to deal with
mixtures. However, it does provide a plausible alternative to local
independence assumptions (especially if you have strong a prior
knowledge about the presence of an edge.)
I hope these ramblings have shed some light on the problem.
--Russell Almond