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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>>
>
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