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Re: [ontolog-forum] Probabilistic Ontologies

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, "John F. Sowa" <sowa@xxxxxxxxxxx>
Cc: ontolog-forum@xxxxxxxxxxxxxxxx
From: Kathryn Blackmond Laskey <klaskey@xxxxxxx>
Date: Mon, 18 Jun 2007 15:33:06 -0400
Message-id: <p0611044fc29c83936453@[]>
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)

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