Hi Ben,
Well, Jaynes showed that the PI can be derived from another assumption, right?: That equivalent states of information yield equivalent probabilities....
Yes, as I understand it the principle of indifference is a special case of Jaynes' principle of maximum entropy.
I have no problem with the principle of maximum entropy except in this special case in which we have *zero* information relevant to the true probabilities of outcomes. The ramifications of mathematical concepts can sometimes change radically when considering zero quantities, and I strongly suspect this is an example.
It seems to me that in this special case of the maximum entropy principle, known as the indifference principle, we are not actually considering 'equivalent states of information' which might in theory yield equivalent probabilities. We are not at all considering information. We have no pieces of information to analyze, test, compare, or otherwise consider. Instead we are considering non-information (whatever that means).
How can we justify probabilistic inferences from non-information? How can we justify a decision to infer something from nothing?
I use the word "justify" here in the formal epistemological/logical sense. This goes to the question of whether the principle of indifference is truly a valid logical principle, in the formal sense of that word, as is maintained by certain people loyal to certain logical interpretations of probability theory.
As you know I believe the PI falls short of that definition -- that it is instead merely a heuristic device -- a bit of semi-religious quasi-logic left over from the essentially defunct classical theory of Laplace. Perhaps someone can convince me otherwise (you came very close, when you answered the wine/water paradox!)
This seems to also be dealt with at the end of Cox's book...
Interesting. I'm tempted to read Cox's book so that you and I can discuss his ideas in more detail here on your list. (I worry that my enthusiasm for this subject is only annoying people on that other discussion list.) Is that something you would like to do? Please let me know!
I'm copying Jef and Stu here, as this is not the first time the principle of maximum entropy has come up in the dialogue. (I don't want you guys to think your thoughts on this subject went ignored or unanswered.)
-gts ----- This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
