ontolog-forum
[Top] [All Lists]

Re: [ontolog-forum] Probabilistic Ontologies

 To: "[ontolog-forum]" Waclaw Kusnierczyk Tue, 19 Jun 2007 08:45:10 +0200 <46777B76.5050001@xxxxxxxxxxx>
 ```Waclaw Kusnierczyk wrote: > Kathryn Blackmond Laskey wrote: >> To do probability right, you need real numbers, and you need to be >> able to do continuous distributions. For example, consider a number >> picked at random between zero and one. (Imagine a spinner that can >> land anywhere on the disk with equal likelihood; measure the distance >> around the edge to where it landed and divide by the circumference -- >> the number you get is uniformly distributed between zero and one.) >> The probability of getting a number in any sub-interval of the unit >> interval is equal to the length of the sub-interval. But the >> probability of getting any specific number is zero. >> >> This raises some tricky mathematical issues. In finite domains, we >> can equate probability zero with unsatisfiability and probability 1 >> with validity. But in the example I gave above, UniformRand = X is >> satisfiable for any X between zero and one, yet (Prob (UniformRand = >> X)) is zero for all X, and moreover, Prob(Exists X UniformRand=X)) >> is equal to 1. > > Isn't the issue due to approximation? In P(X = x) = 0 for any x in the > selected interval of real numbers, what is the zero? The probability > that a random variable will take on a specific value from a particular > set of values is per definitionem equal to the inverse of the > cardinality of the set -- right? An interval -- a dense set of real > numbers, e.g., the unit interval (0.0, 1.0) -- has the cardinality > aleph_0 (I guess). The zero in P(X = x) = 0 is in fact the inverse of > aleph_0; it is not exactly the usual 0, it is an approximation. For > every x in the interval, P(X = x) is infinitesimal, but is not exactly > 0. If infinitesimals were equal to 0, would integration make sense? > It may be practically zero, but when discussing paradoxes of > probability, it may be worth mentioning.    (01) ... thus in any non-empty interval there are aleph_0 real numbers x for each of which P(X = x) is non-zero, and thus obviously P(exists x: X = x) is 1.    (02) > (Ignore if I am getting this wrong.)    (03) Still holds.    (04) vQ    (05) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Subscribe/Config: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (06) ```
 Current Thread Re: [ontolog-forum] Probabilistic Ontologies, (continued) Re: [ontolog-forum] Probabilistic Ontologies, Kathryn Blackmond Laskey Re: [ontolog-forum] Probabilistic Ontologies, John F. Sowa Re: [ontolog-forum] Probabilistic Ontologies, Kathryn Blackmond Laskey Re: [ontolog-forum] Probabilistic Ontologies, John F. Sowa Re: [ontolog-forum] Probabilistic Ontologies, Kathryn Blackmond Laskey Re: [ontolog-forum] Probabilistic Ontologies, Pat Hayes Re: [ontolog-forum] Probabilistic Ontologies, Kathryn Blackmond Laskey Re: [ontolog-forum] Probabilistic Ontologies, Pat Hayes Re: [ontolog-forum] Probabilistic Ontologies, John F. Sowa Re: [ontolog-forum] Probabilistic Ontologies, Waclaw Kusnierczyk Re: [ontolog-forum] Probabilistic Ontologies, Waclaw Kusnierczyk <= Re: [ontolog-forum] Probabilistic Ontologies, Barker, Sean (UK) Re: [ontolog-forum] Probabilistic Ontologies, Waclaw Kusnierczyk Re: [ontolog-forum] Probabilistic Ontologies, Barker, Sean (UK) Re: [ontolog-forum] Probabilistic Ontologies, Waclaw Kusnierczyk Re: [ontolog-forum] Probabilistic Ontologies, Barker, Sean (UK) Re: [ontolog-forum] Probabilistic Ontologies, John F. Sowa Re: [ontolog-forum] Probabilistic Ontologies, John F. Sowa Re: [ontolog-forum] Two ontologies that are inconsistent but both needed, Pat Hayes Re: [ontolog-forum] Two ontologies that are inconsistent but both needed, Kathryn Blackmond Laskey Re: [ontolog-forum] Two ontologies that are inconsistent but both needed, Christopher Menzel