John, (01)
Yes, the kinds of representations that are needed could be encoded as
meta-level statements in IKL. It's more complex than simply
attaching probabilities to logical statements, but it's
straightforward. (02)
Paulo Costa and I recently put together a simple example of pizzas
and wines. See
http://ite.gmu.edu/~klaskey/PizzaWine/PizzaWineExample.xhtml. (03)
Consider for example the Pizza Base MFrag, a fragment of a Bayesian
network that represents the pizza bases a customer is likely to
choose given his/her choice(s) of wine. (04)
We have: (05)
(1) Context random variables: These are constraints that have to be
satisfied for the probability distributions to make sense. They are
IsA(Pizza,p), IsA(Wine,w), and ServedWith(w,p). (06)
(2) Resident random variable: This is a random variable whose
distribution is defined in the MFrag. In this case, we have one
resident random variable, PizzaBase(p), with possible values
ThinAndCrispyBase and DeepPanBase. (07)
(3) Input random variables: These are random variables that influence
the probability of the resident random variable. In this case, the
pizza base is influenced by WineBody(w) and WineFlavor(w). (08)
(4) Local distributions: An instance of the resident random variable
will created for each pizza instance in the situation about which we
are reasoning. To construct its probability distribution, we need a
function that maps the WineBody and WineFlavor of all the wine
instances satisfying the ServedWith relation to a probability
distribution on the values ThinAndCrispyBase and DeepPanBase. (09)
So this MFrag says that the probability distribution for the type of
pizza base a customer is likely to order depends on the body and
flavor of the wine(s) that are served with it. The MFrag defines a
function for mapping wine and body flavors to probability
distributions on pizza bases. (010)
All this could be represented by meta-level statements in IKL. It
would not be too difficult to write an upper ontology for MFrags in
IKL, similarly to the OWL upper ontology we built for PR-OWL. We
might call it PRIKL. :-) (011)
Kathy (012)
At 7:44 AM -0400 6/17/07, John F. Sowa wrote:
>Kathy,
>
>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.
>
>> 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 can be defined in the Horn-clause subset
>of IKL, which is a very efficient superset of OWL.
>
>> Sophisticated probability can be thought of as having two parts:
>> the structural and the numerical. The structural part represents:
>> (1) a set of random variables (uncertain features or relationships);
>> (2) the possible values each random variable can take on); and
>> (3) conditional dependency relationships.
>
>That could be represented in IKL.
>
>> For example, suppose we are trying to identify aircraft using radar
>> reports. Consider two entities, a flying object and a sensor. We
>> have two random variables: ObjectType and SensorReport. The possible
>> values of each of these are {FighterAircraft, OtherAircraft, Bird}.
>> The probability distribution for SensorReport depends on ObjectType.
>
>Could you give a specific example of the representations that are
>currently used (preferably in the usual math notation, not in OWL).
>
>John
>
>
>
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