Hi,
Well, Jaynes showed that the PI can be derived from another
assumption, right?: That equivalent states of information yield
equivalent probabilities....
This seems to also be dealt with at the end of Cox's book "The
Algebra of Probable Inference" where he derives the standard entropy
formula for measuring information from some fundamental axiomatic
assumptions.... One can define information axiomatically in this
manner, then make an assumption of maximum entropy priors, and the PI
follows from that...
But so far as I can tell this still requires an additional assumption
beyond those required in order to derive probability theory...
-- Ben
On Jan 28, 2007, at 1:10 PM, gts wrote:
Hi Ben,
On Extropy-chat, you and I and others were discussing the
foundations of probability theory, in particular the philosophical
controversy surrounding the so-called Principle of Indifference.
Probability theory is of course relevant to AGI because of its
bearing on decision theory (I assume that's why you invited me
here. :)
As you know, the Principle of Indifference (PI) states that if no
reason exists to prefer any of n possibilities then each
possibility should be assigned a probability equal to 1/n. The PI
is known also as the Principle of Insufficient Reason, the name
given it by classical probabilists who followed after Laplace, who
took it for granted as a self-evident principle of logic. (It was
John Maynard Keynes who later renamed it the Principle of
Indifference.)
I found a discussion of the Principle of Insufficient Reason in
this book about decision theory:
Choices: An Introduction to Decision Theory By Michael D. Resnik
http://books.google.com/books?
vid=ISBN0816614407&id=4genrKNUkKcC&pg=RA2-PA35&lpg=RA2-
PA35&ots=wE4Uxk7bqE&dq=principle+of+insufficient
+reason&sig=PsMUy3fqcMgFha8Kyx2HLaC-EA8
This author criticizes the PI in two ways. His first is mainly
philosophical: if there is no reason for assigning one set of
probabilities rather than another, then there is no reason for
assuming the states are equiprobable either. This is pretty much
the same argument I was trying to make on ExI.
His second objection is one we had not discussed: though the PI
seems like a common-sense way to proceed under conditions of
uncertainty, invoking it can sometimes lead to disastrous
consequences for the decision-maker. I would add that while the PI
might be useful in some situations as a heuristic device in
programming AGI, perhaps some accounting should be made for the
extra risk it entails.
-gts
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