A Bayesian Perspective to
Semantic Web – Uncertainty modeling
in OWL
Jyotishman Pathak
04/28/2005
Why did I choose this topic?
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


My research: Semantic Web
ComS 673: Bayesian Network
Rendezvous between BN & SW
References


A Bayesian Approach to Ontology in OWL Ontology, Zhongli Ding
et al., In Proc. of AISTA-2004
A Probabilistic Extension to Ontology Language OWL, Zhongli
Ding et al., In Proc. of HICSS-2004
http://www.csee.umbc.edu/~zding1
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Spring-2005 CS-673 Final Project
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
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Preliminaries – Semantic Web for Dummies!
Semantic Web
The book
does not
really exist!
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Preliminaries – Semantic Web (1)

Current Web Architecture



Network of hyper links
O.K. for human-processing (e.g., Natural Language,
Graphics)
Difficult for machine processing (ambiguity,
unconstrained data formats)
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Preliminaries – Semantic Web (2)
Do
like
Do you
you like
Golf?
Golf?
Do you like
Golf?
No. I prefer
Mustang
• Same term, different meaning
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Preliminaries – Semantic Web (3)

The Semantic Web is an extension of the current
web that will allow you to find, share, and combine
information more easily.

Extend the current web (do NOT define a new one!)

Express information in a format that is:



Unambiguous
Amenable to machine processing
Add metadata (to describe existing or new data)
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Preliminaries – Semantic Web (4)
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An Ontology is an engineering artifact:


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Describes formal specification & shared understanding
of a certain domain
Formal and machine manipulable model of the domain
Decades of research done by KR community
Ontologies have two main components:

Names for important concepts in the domain


Elephant is a concept whose members are a kind of Animal
Background knowledge/constraints on the domain
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Every Elephant is either an African_Elephant or an
Indian_Elephant
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Preliminaries – Semantic Web (5)

OWL: Web Ontology Language (W3C Recommendation)
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Is written using XML-based syntax
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Categorizes the basic concepts in terms of Classes:
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classes can be viewed as “sets” of possible concepts
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E.g., Animal in our example
hierarchies of concepts can be defined as sub-classes
Union, Intersection, Disjoint, Complement etc..
Properties are defined by:
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constraints on their range and domain, or
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
E.g., type of the Elephant can be either African or Indian
specialization (sub-properties)
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Property
Domain
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Range
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<owl:Class rdf:ID="Vegetarian">
<rdfs:subClassOf rdf:resource="http://xmlns.com/foaf/0.1/#Person"/>
<rdfs:subClassOf>
<owl:Restriction>
<owl:onProperty rdf:resource="#eats"/>
<owl:allValuesFrom rdf:resource="#VegetarianFood"/>
</owl:Restriction>
</rdfs:subClassOf>
Person
</owl:Class>
subClass
<owl:Class rdf:ID="Vegan">
<rdfs:subClassOf rdf:resource="#Vegetarian"/>
<rdfs:subClassOf>
<owl:Restriction>
<owl:onProperty rdf:resource="#eats"/>
<owl:allValuesFrom rdf:resource="#VeganFood"/>
</owl:Restriction>
</rdfs:subClassOf>
</owl:Class>
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Vegetarian
subClass
Vegan
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
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Introduction and Motivation - I

OWL allows us to define classes, properties etc.
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Unfortunately, OWL is based on crisp logic



A vegan only eats vegan food
An elephant can be either African or Indian
Real life (data) has uncertainty associated
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Introduction and Motivation - II

Uncertainty in Ontology Representation

Degree of Inclusion


Besides A subclassOf B, also A is a small subset of B
Degree of Overlap (Intersection)

A
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A and B overlap, but none is a subclass of the other
A
BB
B
A
A
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B
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Introduction and Motivation - III
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Uncertainty in Ontology Mapping


Similarity between concepts in different ontologies
cannot be adequately represented by logical relations
Mappings are hardly 1-to-1
A
A’
A B’
B
subClass
C
B
subClass
B’
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Similar /
Equivalent
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subClass
C
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Introduction and Motivation - IV
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Thus,
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Existing logic based approaches are inadequate to
model Ontological uncertainty
Uncertainty is more prevalent in presence of multiple
Ontologies
Reasoning becomes a problem
Leverage on approaches for graphical models
This work builds on Bayesian Network. Why?
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Structural similarity between the DAG of a BN and the
graph of OWL ontology
BN semantics is compatible with that of OWL
Rich set of efficient algorithms for probabilistic
reasoning and learning
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Overview of Uncertainty Modeling in
Ontology
Onto
Probabilistic
annotation


Reasoning
OWL-BN
translation
Not supported by current OWL
Define new classes for prior and conditional probabilities
Structural Translation



BN
Encoding Probabilities in Ontology


P-Onto
Class hierarchy: set theoretic approach
Logical relations (equivalence, complement, disjoint, union,
intersection): introducing control nodes
Constructing CPTs

Decomposed Iterative Proportional Fitting Procedure (D-IPFP)
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
04/28/2005
Spring-2005 CS-673 Final Project
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Encoding Probabilities in Ontology - I


Two kinds of probabilistic information

Prior or marginal probability P(C);

Conditional probability P(C|OC), where OCC,
OC≠.
C≠,
Three new OWL classes: “PriorProb”, “CondProb”,
“Variable”


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PriorProb: “hasVariable”, “hasProbValue”
CondProb: “hasCondition” (1 or more), “hasVariable”,
“hasProbValue”
Variable: “hasClass”, “hasState”
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Encoding Probabilities in Ontology - II

Example 1: P(c) = 0.8
<Variable rdf:ID="c">
<hasClass>C</hasClass>
<hasState>True</hasState>
</Variable>
<PriorProb rdf:ID="P(c)">
<hasVariable>c</hasVariable>
<hasProbValue>0.8</hasProbValue>
</PriorProb>

Example 2: P(c|p1,p2,p3) = 0.8
<Variable rdf:ID="c">
<hasClass>C</hasClass>
<hasState>True</hasState>
</Variable>
<Variable rdf:ID="p1">
<hasClass>P1</hasClass>
<hasState>True</hasState>
</Variable>
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<Variable rdf:ID="p2">
<hasClass>P2</hasClass>
<hasState>True</hasState>
</Variable>
<Variable rdf:ID="p3">
<hasClass>P3</hasClass>
<hasState>True</hasState>
</Variable>
<CondProb rdf:ID="P(c|p1, p2, p3)">
<hasCondition>p1</hasCondition>
<hasCondition>p2</hasCondition>
<hasCondition>p3</hasCondition>
<hasVariable>c</hasVariable>
<hasProbValue>0.8</hasProbValue>
</CondProb>
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
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Structural Translation - I
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Every primitive or defined concept class C, is mapped
into a two-state (either “True” or “False”) variable node in
the translated BN;
There is a directed arc from a parent superclass node to
a child subclass node;
C is true when an instance
x belongs to it
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Structural Translation - II
Control Nodes
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Structural Translation - III
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
04/28/2005
Spring-2005 CS-673 Final Project
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Constructing CPTs
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Two kinds of nodes:

XC: control nodes for bridging nodes which are
associated by logical relations

XR: regular nodes for concept classes

P(C) or P(C|OC), where OCC, C≠, OC≠

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Initially assigned Prior or Conditional probabilities in the
OWL file
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CPTs for Control Nodes
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CPT for Regular Nodes
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CT: the situation in which all the control nodes in
BN are “True”

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Logical relations defined in original Ontology are held
in the translated BN
Goal: To construct CPT’s for regular nodes in XR,
such that P(XR | CT) is consistent with initial
constraints
Problem:

Constraints not given in the form of CPT
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P(C | A, B) vs. P(C | A)
We cannot determine CPT for node C directly
CPT
Constraint
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CPTs for Regular Nodes - Method

Solution:


Decomposed Iterative Proportional Fitting
Procedure (D-IPFP)
IPFP: a well-known mathematical procedure
that modifies a given distribution to meet a set
of constraints while minimizing I-divergence to
the original distribution
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CPTs for Regular Nodes - I-divergence
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CPTs for Regular Nodes - I-projection
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CPTs for Regular Nodes - IPFP
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CPTs for Regular Nodes - D-IPFP
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Example - I
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Example - II
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
04/28/2005
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Reasoning
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


Concept Satisfiability: P(e | CT )  0 ?
Concept Overlapping: P(c | e, CT ) = ?
Concept Subsumption
…
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Outline

Preliminaries

Semantic Web & related concepts

Motivation

Translating OWL Taxonomy to BN
 Encoding Probabilities in Ontology
 Structural Translation
 Constructing CPTs

Reasoning

Conclusion
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Conclusion

Summary
A principled approach to uncertainty modeling in
ontology
 do
Ontology
mapping
 Allows us to
reasoning
in presence of partial
 A parsimonious set of links
knowledge
onto1
onto2
 Capture similarity between
concepts by joint distribution
 Can be used successfully
for Multi-Ontology
Mapping
 Mapping as evidential
reasoning
Probabilistic
Probabilistic

ontological
information
P-onto1
 Current
Probabilistic
annotation

BayesOWL: Probabilistic
Framework for Uncertainty in
Semantic Web
P-onto2
ontological
information
work (as of Summer-2004)
BN1
Prototype development
BN2
OWL-BN
 Experimentation with real world Ontologies
translation

concept
mapping
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