Pei,
I wonder if Cox's Assumption 1 could be salvaged by replacing it
with, say, an assumption that
"Among a set of statements supported by equivalent amounts of
evidence, the relative plausibility of an individual
statement may be assessed by a single real number."
Based on this modified assumption, I think a variation on Cox's
arguments could probably be made to work.
Would this modification address your objection?
-- Ben
On Feb 2, 2007, at 4:13 PM, Pei Wang wrote:
Ben,
To me, not only Assumption 3 is too strong, but also Assumption 1,
which does assume that a real number is enough for the "plausibility
of a statement". For this reason, these assumptions do not even "holds
approximately" in the AGI context --- using one number or two numbers
makes a huge difference, which I'm sure you know well.
The Halpern vs. Snow debate is largely irrelevant to this issue. I
mentioned them just to show that Cox's work is well known to the UAI
community.
Pei
On 2/2/07, Ben Goertzel <[EMAIL PROTECTED]> wrote:
The paper Pei forwarded claims that Cox's arguments don't work for
the discrete case, but the attached paper from Snow in 2002 [which
will come through if this listserver allows attachments...] presents
a counterargument, suggesting that a variant of Cox's argument does
in fact work for the discrete case.
However, my contention is that Cox's assumptions, while reasonable,
are too strong to be viably assumed for a finite-resources AI system
(or a human brain).
To see why, look at Assumption 3 in
http://en.wikipedia.org/wiki/Cox's_theorem
which states basically that
"
Suppose [A & B] is equivalent to [C & D]. If we acquire new
information A and then acquire further new information B, and update
all probabilities each time, the updated probabilities will be the
same as if we had first acquired new information C and then acquired
further new information D.
"
This is not exactly the case in Novamente, nor in the human brain.
So one question is: If this assumption holds only to approximately in
an AI system (or other mind), how inaccurate is the ensuing
approximation of probabilistic correctness constituted by its
judgments? I.e., how wide are the error bars on the conclusion of
Cox's Theorem, when its assumptions are approximately varied?
-- Ben
On Feb 2, 2007, at 2:39 PM, Pei Wang wrote:
>> > I don't know of any work explicitly addressing this sort of
>> issue, do
>> > you?
>>
>> No, none that address Cox and AI directly, but I suspect one is
>> forthcoming perhaps from you. Yes? :)
>
> There is a literature on Cox and AI. For example,
> http://www.cs.cornell.edu/home/halpern/papers/cox1.pdf
>
> Pei
>
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