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Re: [ontolog-forum] Probabilistic Ontologies

 To: "[ontolog-forum] " , "John F. Sowa" ontolog-forum@xxxxxxxxxxxxxxxx Kathryn Blackmond Laskey Mon, 18 Jun 2007 15:33:06 -0400
 ```Pat,    (01) > >What I was asking is how a language such as IKL, which is a >>superset of FOL that also supports metalevel statements, could >>be used to represent the kinds of operations required for >>probability models. >> >>> What you describe is far too simplistic. It's nearly impossible >>> to create a probability model that way that's not either utterly >>> simplistic or inconsistent. >> >>I used a very simple example, but the IKL mechanisms can be used >>to support metalevel statements about propositions, the structural >>components of propositions, their relationships to numerical >>values, and the operations on those values. > >IKL supplies natural numbers but it doesnt have built-in reals. So >you will need some extra machinery of some kind to be able to do >continuous math, for example for talking about distribution curves or >integrals.    (02) That's correct.    (03) 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.    (04) 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.    (05) This problem comes up in countably infinite domains, too. For example, consider infinitely many tosses of a fair coin. Any sequence of heads and tails is possible, but each sequence has probability zero. To be consistent with the statistical theory, the sequence has to have 50% heads and 50% tails (to be precise, the limit as n approaches infinity of the number of heads in the first n tosses divided by n has to be 0.5). Sequences with a different limiting proportion, or with no limiting proportion, are inconsistent with the statistical theory.    (06) > >> Over the past several decades, statisticians and computer >>> scientists have learned a great deal about how to represent >>> probabilistic knowledge. >> > >I'm sure they have, but the IKL mechanisms can support those >>representations. Anything that can be defined in PR-OWL or > >BayesOWL can be defined in IKL plus much, much more. Numerical >>functions of any kind > >Any kind??    (07) If we restrict ourselves to IKL functions that can be represented using a finite number of bits, there are only countably many of them. There are uncountably many probability distributions on the integers.    (08) Kathy    (09) _________________________________________________________________ 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    (010) ```
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