Representing Probabilistic Knowledge
in the Semantic Web

To Describe Propositions

Proposition here refers to a predicate with zero or more arguments (and modifiers). It is different from statement in that we cannot say if the content is true or not, so it is virtual or infinite in a sense.

A proposition is expressed by a blank node of type prob:Clause. A node of type prob:Clause usually takes a property prob:Predicate with a blank node of type rdf:Property as its value, which corresponds to the predicate of the proposition and zero or more properties that correspond to the arguments and modifiers.

The following is an example of the description in N3.

[	a	prob:Clause;
	prob:predicate	ex:p;
	role:foo	ex:r1;
	  ...
	role:bar	ex:r2
].

To Describe Negations

To describe that a proposition X is a negation of a proposition Y, i.e. one is TRUE whenever the other is FALSE, create a prob:Clause node x whose prob:negationOf is a prob:Clause y.

_:x	a	prob:Clause;
	prob:nagationOf	_:y.

To Describe Unconditional Probabilities

To describe that a proposition X holds with probability P, create a prob:ProbabilisticStatement node whose prob:consequence is x and prob:hasProbability is p, where x is an instance of prob:Clause representing X and p is an instance of prob:Probability representing P.

[	a	prob:ProbabilisticStatement;
	prob:consequence	x;
	prob:hasProbability	p
].

To Describe Conditional Probabilities

To describe that a proposition X holds with probability P given a set of premises {Y₁, ..., Y_n}, create a prob:ProbabilisticStatement node that has prob:consequence x, prob:condition y_1, ..., prob:condition y_n, and prob:hasProbability is p, where x,y_1, ..., y_n are instances of prob:Clause representing X and Y₁, ..., Y_n respectively and p is an instance of prob:Probability representing P.

[	a	prob:ProbabilisticStatement;
	prob:consequence	x;
	prob:condition	y_1;
	...
	prob:condition	y_n;
	prob:hasProbability	p
].

To Describe Observations

To describe that a proposition O is observed with probability P, create a prob:Observation node whose prob:proposition is o and prob:hasProbability is p, where o is an instance of prob:Clause representing O and p is an instance of prob:Probability representing P.

[	a	prob:Observation;
	prob:proposition	o;
	prob:hasProbability	p
].

To Describe Posteriors

To describe that a proposition X is concluded to hold with probability P, create a prob:Belief node whose prob:consequence is x, and prob:hasProbability is p, accompanied by y's for each observation with label prob:observation.
Note that the observation node is of type prob:Observation, not prob:Clause.

[	a	prob:Belief;
	prob:consequence	x;
	prob:observation	y;
	prob:hasProbability	p
].

Framework

As a use case of the vocabulary, I'd like to present a framework for probability calculations using RDF and Bayesian network, .

1. Describe the problem by a RDF graph using above proposed vocabulary
2. Convert the graph into a Bayesian network, and export it to a Bayesian network store.
3. Describe the observation by a RDF graph
4. Convert the observation graph into a query for a Bayesian reasoner on the store and hand it to the reasoner.
5. Do the calculations on the reasoner and import the result back to the RDF store and merge.

Bayesian Network

A Bayesian network is a directed acyclic graph (DAG), representing probabilistic dependencies among values of the variables. The nodes and links represent the variables, and the causal relationship between them, respectively. Each node is accompanied with a conditional probability table (CPT) that represents the probabilistic relationship between the variables. The posterior probability distributions ("beliefs")for each variable could be calculated by propagating beliefs. For an example see the example below.

When exporting Bayesian networks, the XMLBIF format is used here. The XMLBIF is a XML-based format for exchanging Bayesian networks, proposed by Fabio Cozamn et al.

Corresponding Bayesian Network

This is the Bayesian network for the metastatic cancer case. Its serialization in the XMLBIF is here. The XML fragments concerning the HeadAche node are, for example, as following.

In XMLBIF, nodes and accompanying CPTs in a Bayesian network are represented by VARIABLE and DEFINITION elements respectively.

<VARIABLE TYPE="nature">
	<NAME>ex:hasSevereHeadAche</NAME>
	<OUTCOME>TRUE</OUTCOME>
	<OUTCOME>FALSE</OUTCOME>
</VARIABLE>
<DEFINITION>
	<FOR>ex:hasSevereHeadAche</FOR>
	<GIVEN>ex:hasBrainTumor</GIVEN>
	<TABLE>0.8 0.2 0.6 0.4</TABLE>
</DEFINITION>

Observations

The graph representing the observation that the patient has severe headaches but is not in coma, is represented by the following graph. This should be handed to the Bayesian network reasoner in an appropriate form.

[ a	prob:Observation;
	prob:proposition	_:HeadAche;
	prob:hasProbability	[a prob:Probability;
				 rdf:value	1.0]
].

[ a	prob:Observation;
	prob:proposition	_:NotComa
	prob:hasProbability	[a prob:Probability;
				 rdf:value	1.0]
].

Probability Calculation

The Bayesian network above is "multiply-connected", i.e. there is one or more "loop" when seeing the links undirected, so the calculation (propagation) of the probability needs some technique (Junction Tree, Cutset Conditioning, Sampling...). As the result of the calculation, we get P(MetastaticCancer | HeadAche & !Coma) = 0.097.

Importing Back the Result

The graph representing the result of the calculation is as follows. It should be merged to the graph representing the knowledge along with those for the observations.

[	a	prob:Belief;
	prob:consequence	_:MetastaticCancer;
	prob:observation	_:ObsHeadAche;
	prob:observation	_:ObsNotComa;
	prob:hasProbability	[a	prob:Probability;
				 rdf:value	0.097]
].