Judea,
1. I'll let Philippe respond regarding TBMs.
2. I agree that the lower envelope picture has problems with conditioning
if you use Dempster's rule. There is another rule that works fine
and gives perfectly reasonable answers under the lower envelope
interpretation. (The rule is that Bel(A|B) = Bel(A \inter B)/Bel(A
\inter B) + Pl(A^c \inter B). As Ron Fagin and I, and independently
Jean-Yves Jaffray, showed, this gives a belief function that is the
lower envelope of the result of conditioning the family that generated
Bel in the first place.) The lower envelope interpretation does lose
information. One consequence of this, as Ron and I showed (perhaps
that's what you're referring to) is that updates don't commute. My
feeling is that there's no real point in using belief function as lower
envelopes -- you're better off working directly with the underlying
family of distributions -- but it is a perfectly coherent interpretation.
Out of curiosity, have you ever considered how updating should work in
your probability of provability interpretation? Is it obvious that
Dempster's Rule is the appropriate updating rule under that
interpretation?
3. I don't know what qualifies as a "canonical" example, but Ron and I
certainly gave examples in our paper of how one could quantify "belief
as evidence". The basic pictures is that you're assumed to start with a
set of hypotheses (e.g., about the probability that a coin will land
heads) and observations. You want to compute the belief (and
plausibility) that you assign to hypothesis H given some observations.
(E.g., what is your belief that a coin is fair given that you've seen 7
heads? I know a Bayesian would just start with a prior and condition,
but this is a non-Bayesian approach.) We instantiated this
with a coin that is known to be either biased towards heads (with
probablity 2/3 of landing heads) or biased towards tails (with a
probability 2/3 of landing tails), but the general framework certainly
applies whenever you have hypotheses and observations and you want to
assign beliefs to hypotheses. (Of course, you may have to pound your
application a bit before it fits into this framework.) To be fair, Ron
and I -- and Peter Walley before us -- also showed that if you want
certain other desiderata, this interpretation is best captured by a
belief function that also happens to be a probability function, so again
there's no compelling reason to use belief functions here (although
Dempster's Rule works fine).
Bottom line: I think there really are at least three quite different
interpretations of belief, all of which have been used in the literature
(not always coherently and not just in my papers). I suspect there are
others as well.
-- Joe
Joe Halpern wrote:
Judea's probability of provability is one, but only one
among several. Another one is Philippe's TBM. Yet another is belief
functions as lower envelopes of families of probability ..
Joe,
I will be the first to buy your "thousand-flower" model
as soon as I find two flowers blooming. So far, I have found only one.
1. TBM is still waiting for canonical example
to convince us that one need to quantify the 'strength
of ones opinion", and in a TBM-specific way.
2. The "lower envelop" picture does not work --
and I believe you and Fagin showed it -- as soon
as we start conditioning on some evidence, the
envelop ceases to be a belief-function.
3. Belief as evidence ? same as TBM; waiting for canonical example
(and I read your paper) where it is clear that what we
need to quantify is the "evidence for a proposition",
rather than the strength of one's belief in a proposition.
So far we have one flower and two contending linguistic expressions:
"strength of opinion" and "belief as evidence",
and with this ammunition we need to convince Kathy that
sometimes it is important
to compute BF rather than probabilities.
Recall, Kathy does not fall for just fancy expressions,
she requested a SITUATION with compelling REASON why it is
CORRECT to compute one quantity and INCORRECT to compute
another.
Give me two, and I will buy the thousand.
=========Judea
