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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 08:45:10 +0200
Message-id: <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)

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