Pei, I finally took a moment to actually read your email...
> > However, the negative evidence of one conclusion is no evidence of the > other conclusion. For example, "Swallows are birds" and "Swallows are > NOT swimmers" suggests "Birds are NOT swimmers", but says nothing > about whether "Swimmers are birds". > > Now I wonder if PLN shows a similar asymmetry in induction/abduction > on negative evidence. If it does, then how can that effect come out of > a symmetric truth-function? If it doesn't, how can you justify the > conclusion, which looks counter-intuitive? According to Bayes rule, P(bird | swimmer) P(swimmer) = P(swimmer | bird) P(bird) So, in PLN, evidence for P(bird | swimmer) will also count as evidence for P(swimmer | bird), though potentially with a different weighting attached to each piece of evidence If P(bird) = P(swimmer) is assumed, then each piece of evidence for each of the two conditional probabilities, will count for the other one symmetrically. The intuition here is the standard Bayesian one. Suppose you know there are 10000 things in the universe, and 1000 swimmers. Then if you find out that swallows are not swimmers ... then, unless you think there are zero swallows, this does affect P(bird | swimmer). For instance, suppose you think there are 10 swallows and 100 birds. Then, if you know for sure that swallows are not swimmers, and you have no other info but the above, your estimate of P(bird|swimmer) should decrease... because of the 1000 swimmers, you now know there are only 990 that might be birds ... whereas before you thought there were 1000 that might be birds. And the same sort of reasoning holds for **any** probability distribution you place on the number of things in the universe, the number of swimmers, the number of birds, the number of swallows. It doesn't matter what assumption you make, whether you look at n'th order pdf's or whatever ... the same reasoning works... >From what I understand, your philosophical view is that it's somehow wrong for a mind to make some assumption about the pdf underlying the world around it? Is that correct? If so I don't agree with this... I think this kind of assumption is just part of the "inductive bias" with which a mind approaches the world. The human mind may well have particular pdf's for stuff like birds and trees wired into it, as we evolved to deal with these things. But that's not really the point. The inductive bias may be much more abstract -- ultimately, it can just be an "occam bias" that biases the mind to prior distributions (over the space of procedures for generating prior distributions for handling specific cases) that are simplest according to some wired-in simplicity measure.... So again we get back to basic differences in philosophy... -- Ben G ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=114414975-3c8e69 Powered by Listbox: http://www.listbox.com
