Your belief in A increases, but you don't think your belief in (not A) should
decrease? [...] such a position seems inconsistent with deductive logic [...]
[...] any evidence in favor of A is evidence against (not A). To increase your
belief in A without decreasing your belief in (not A) could only be justified
if
you were willing to admit the possibility that both A and (not A) could be
simultaneously true, in contradiction to deductive logic.
Smets Philippe replied:
I wonder if you ever read about Dempster Shafer theory, about the
transferable belief model, and about Possibility theory (if you need
references, ask for it, I'll send you what you would like).
No, I haven't studied Dempster Shafer theory in any detail; my emphasis has been
on Bayesian methods. (And yes, please send me references.) But, from what I do
know about Dempster Shafer theory, it is irrelevant in the context of this
discussion, which began with Cox's Theorem and your objection to the assumption
that the plausibility or belief in (not A) should be a function of the
plausibility or belief in A. You stated that an increase in your belief in A
shouldn't require a decrease in your belief in (not A).
Now, an explicit assumption of Cox's Theorem is that degrees of belief or
plausibility are one-dimensional, represented by a single number, and it is in
this context that the requirement that Bel(not A) be a function of Bel(A) must
be understood. If I understand correctly, Dempster Shafer theory uses a
two-dimensional belief representation, with one interpretation of the two
numbers being upper and lower bounds on the (frequentist) probability of the
proposition in question. Thus it would seem that you and I are using the same
term ("degree of belief") for two entirely different concepts, which is why I
see contradictions where you see none. I stand by my comments for theories that
represent belief or plausibility of a proposition by a single number; I have no
opinion for multi-dimensional theories, as I have not yet studied them in any
detail.
> And in my theory, called the transferable belief model, if bel(A) = 1, then
> indeed bel(notA) = 0. But this does not means that bel(notA) = 0 implies
> bel(A) = 1.
Here again we see a large conceptual difference in what is meant by "belief".
In the context of Cox's Theorem, it is assumed that there are degrees of belief
(which need not be assumed to be 0 and 1) that correspond to "certainty of
falsity" and "certainty of truth". In this context, if the degree of belief in
a known false proposition is F (= 0 in your example), then any proposition with
a degree of belief F is, by definition, known to be false, and so bel(not A) = F
must indeed imply that A is known to be true. None of this discussion, by the
way, presupposes probability theory.
In light of this discussion, would it be accurate to say that your real
objection is to the use of a one-dimensional notion of belief instead of
two-dimensional beliefs?
I have not been motivated to learn more about Dempster Shafer theory and
variants for the following reasons:
1) I see no need for the added complication introduced by D-S. What problem is
being solved by this added complexity? Can you point to pathologies of Bayesian
methods that are avoided by D-S or similar theories? Can you point to cases
where D-S or similar methods give better results than properly applied Bayesian
methods?
2) The motivation for D-S seems rooted in what Jaynes calls the "Mind Projection
Fallacy", i.e., the assumption that internal constructs of our own minds are
actual physical properties of the external world. I have argued previously on
this list that probabilities are *not* physical properties of any sort -- they
merely encode our incomplete state of knowledge. Yet Shafer' used the following
postulates:
Postulate 1: *Chance* is the limit of the proportion of "positive" outcomes
among
all outcomes.
Postulate 2: Chances, if known, should be used as *belief functions*.
Postulate 1 is a frequentist view of probability (or "chance"). But the limit
mentioned is a meaningless concept. It can never be measured, and the only way
you could assign it any real meaning is if it were some hidden physical
parameter. (I have previously argued in some detail that it is not.) Postulate
2 worries about whether one can know chances, which again presumes that they are
real, physical quantities. If one views probabilities as encoding a state of
knowledge, it is absurd to speak of them as being "unknown", or placing upper or
lower bounds on them, as if one were unsure of one's own state of knowledge.
Of course, I am willing to give any proponent of D-S or similar theories a fair
shot at convincing me that it's worth my time to study them :-)
-- Kevin S. Van Horn