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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