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Re: [ontolog-forum] Probabilistic Ontologies

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Waclaw Kusnierczyk <Waclaw.Marcin.Kusnierczyk@xxxxxxxxxxx>
Date: Tue, 19 Jun 2007 10:43:47 +0200
Message-id: <46779743.8050105@xxxxxxxxxxx>
Barker, Sean (UK) wrote:
> Waclaw
>       From the point of view of practical mathematics, probability
> over continuous distributions is defined through probability density
> functions, with the probability for any interval being the integral of
> the pdf over that interval. If you reduce the interval to zero, then the
> probability goes to zero. No problem there.    (01)

Depends on what you mean by 'reduce to zero'.  There is the empty 
interval, and infinitely many singleton intervals [x,x].  For the empty 
interval, obviously P(X = x | x in the empty interval) = 0.  For any 
singleton interval, it is not that obvious to me that P(X = x) = 0 for 
some x for which the 'reduced to zero' interval [x,x] is considered.    (02)

>       I'm not sure what you want to say with "P(exists x: > X=x) = 1"
> since the domain of X is fixed by definition, and therefore "the
> probability that X takes a value in its domain" doesn't seem to mean
> very much - probability is about random events, not things that are true
> by definition.     (03)

This part comes from Kathy's post, and she should be asked:    (04)

[KL] 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)

vQ    (06)

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