Joe Halpern wrote,
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,
We seem to agree on two major points:
1. If we want to compute the strength of our belief
in the truth of proposition A (or in the occurence of
event A) and if we have all the information to compute
P(A), then we equate our belief
with P(A) and do not bother with Bel(A).
2. If we want to compute P(A) and we do not have
all the information for evaluating P(A), then using
Bel(A) as a surrogate is a bad move, especially
when used with Dempster rule.
Our disagreements surface in cases where
we want to compute a quantity Q
that represents some other mental attitude toward A, not
exactly our belief in the truth of A, but something related
to it, say "the strength of the evidence we have for A"
or "the validity of argumets we heard for A"
or (quoting Philippe) "my opinion about A"
or " the likelihood that A is inevitable, or legal, or
authenticated by authorities", etc. etc.
The questions in such cases are:
3.0 Should we really use the word "belief" to denote Q?
or will it be confusing to unsuspecting students of AI?
3.1 Should we use Bel(A) to evaluate our desired Q?
3.2 Is it really important to compute Q in the first place?
or are we just doing it to ensure that
a beautiful mathematical object like the D-S theory
is not left without some belief-related interpretation?
My attempts at answering 3.1 and 3.2 have led to one
interpretation (i.e., the likelihood that A is inevitable)
with which I can convince students, without hand waving
and with common intuition, that the quantity Q we wish
to compute is
(i) sometimes important,
(ii) requires Bel(A) and not P(A)
Other candidate interpretations which I have seen
thus far do not answer questions 3.1 and 3.2.,
but if people find them satisfying -- so be it.
The point I wished to stress in this forum
is that we do not lose sight of questions 3.0- 3.2
and that we do not forget points 1 and 2.
========Judea
