Marcus et al - >1) I think a philosophically more useful picture of the Universe >than that something "happens" is that the 4-dimensional space-time >Universe just "is" (like an archived movie roll, rather than a >life performance).
And exactly how "just is" any more useful or satisfying than "just happens"? >2) Classical probability theory does not allow this, and indeed >provides no sound interpretation of probabilities. Roger that!!! > >Naively the probability of an event E is p = lim_n->oo[k_n(E)/n], >where k_n(E) is the number of occurrences of event E in the first >n i.i.d. trials. The problem is that the limit may be anything or >not even exist: e.g. a fair coin can give: head, head, head, head, >... i.e. p=1. Of course, this sequence is "unlikely". For a fair >coin, we have p=1/2 with "high probability". But now we have to >explain what "high probability" means. Circularity! > >Actually, there is a way out of this (near but not exact) >circularity. We have reduced the meaning of "probability" to the >meaning of "high probability", which for infinite sequences >corresponds to "probability 1". One could interpret the mathematical >statement "An event that happens with probability 1" as that the >event happens for sure in our Universe. That's (the complement of) >Cournot's principle: An event (singled out in advance) with >probability zero will not happen. > >3) The third interpretation of probabilities as betting ratios >leads us (necessarily?) back to subjective probabilities. If Your subjective beliefs are probabilistic on a sequence of events for which You receive outcome feedback, then You believe with probability 1 that p = lim_n->oo[k_n(E)/n]. Not only that, you believe the same thing about any subsequence of events picked out in advance, and the events don't have to be iid. That is, if ks_ns(E) is the number of occurrences of E in the first ns trials that fall in a given pre-selected subsequence, and ps is the average of the probabilities You assigned to the events in the subsequence, then You believe with probability 1 that ps = lim_ns->oo[ks_ns(E)/n]. In other words, a Bayesian believes with probability 1 that limiting frequencies in any pre-selected subsequence on which (s)he receives outcome feedback will converge to a stable limit that will match his/her limiting average probabilities for the subsequence. So being led back to subjective probabilities leads us full circle back to limiting frequencies. >4) I would appreciate a discussion of my argument that the >*belief* in a truly random universe is "unscientific", >independent of whether it's actually true or not. The argument was >that to claim that a process is random, means claiming that the >observation cannot be perfectly predicted, and claiming that >something can principally not be predicted=explained is an >unscientific statement. I think it's unscientific to be ABSOLUTELY SURE that every process CAN be perfectly predicted/explained. A scientist has an open mind to the possibility that his/her pet theory may be wrong. A scientist is willing to be overturned by empirical data. A dyed in the wool subjective Bayesian who never assigns probability zero to any physically possible event but who truly believes his/her subjective probability assessments will never be convinced by any amount of evidence that her model is wrong. No matter how disconfirming the observations, she will simply think she has seen an incredibly unusual sequence. Kathy _______________________________________________ uai mailing list uai@ENGR.ORST.EDU https://secure.engr.oregonstate.edu/mailman/listinfo/uai
