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Re: [ontolog-forum] Probabilistic Ontologies

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Kathryn Blackmond Laskey <klaskey@xxxxxxx>
Date: Mon, 18 Jun 2007 00:36:56 -0400
Message-id: <p0611041ec29b366a566b@[]>
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:
>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).
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