Brad said, responding to Moshe: > > We have insufficient knowledge, so we need to make some assumptions to > > approximate P(Xi|Xj). I argue that under these circumstances, the best > > assumption to make is that Xi and Xj are independent, (ie, > P(Xi|Xj)=P(Xi)). > > Does this clarify things? > > > You are basically saying, for each unknown P(Xi|Xj), assume it > equals P(Xi). >
One can do better than such a simplistic form of independence assumption though... For instance we have P(C|A) = P(B|A) * P(C|B) + (1- P(B|A))*(P(C)-P(B)* P(C|B))/(1-P(B)) if we assume that * (A intersect B) and (C intersect B) are independent [the first term] * (A intersect ~B) and (C intersect ~B) are independent [the second term] So if you know P(B|A) and P(C|B) then you can guess P(C|A) if you're willing to assume A and C are independent in B and in ~B, but not universally independent. This is basically the PTL "probabilistic deduction rule." So one reasonable heuristic inference strategy is to prefer trains of inference where this kind of "localized independence" can most plausibly be assumed at each step along the way. But even so, after a lot of inference steps, these independence assumptions can sometimes (not always) lead to substantial error. As noted, the human brain incurs substantial error when doing this sort of reasoning, and this could be because it employs similar heuristic independence assumptions, which can cause problems when applied iteratively in the large scale. -- Ben G ------- To unsubscribe, change your address, or temporarily deactivate your subscription, please go to http://v2.listbox.com/member/[EMAIL PROTECTED]
