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### First Order Bayesian Network

Ref: Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA

Literature Review

Presented By:

Syeda Saleha Raza

Faculty of Computer Science

IBA, Karachi

Reasoning in AI

- Reasoning is ability to do inferences.
- Automated reasoning is computing system that helps in doing this.
- Reasoning is one those several characteristics that distinguish humans from machines.
- AI progresses with the vision of having machines either mimicking human behavior OR assisting them in exhibiting those behaviors

Traditional approaches of reasoning

- Logic programs – Deterministic
- Bayesian networks - Probabilistic

Limitations of Bayesian Networs

- Ability to create multiple instances of same node
- Effective mechanism to specify frequency distribution for a node having unbounded number of parents
- Ability to quantify (existentially & universally) over unbounded and potentially infinite number of parents for a given node
- Ability to consider size of population that is not part of domain but is known to exist

Proposition Logic vs Predicate Logic

- Davis (1990) defines

“Logic is a schema for defining languages to describe and reason about entities in different domains of application.”

This is expressed in form of sequence of facts and rules.

- Logic is propositional if it talks about particular instances of entities and their relationships.

Person(John)

Person(Jim)

Person(Jack)

Father(Jim, John), Father(Jack, Jim) -> GrandFather (Jack, John)

- Predicate Logic, also called First Order Logic, can reason about general properties and relationships that apply to collection of individual.

Person(X)

Person(X), Person(Y), Person(Y), Father(X,Y), Father(Z, X) -> GrandFather (Z, Y)

Limitations of FOL

- Systems based on FOL have the ability to represent entities of different types interacting with each other in varied ways
- However systems based on FOL lack theoretically principled, widely accepted, logically coherent methodology for reasoning under uncertainty

Bayesian Network vs First Order Logic

- Bayesian Networks are probabilistic in nature but represents one particular instance of problem.
- First Order logic is generalization for all instances but deterministic in nature
- The strength of one is the weakness of other and vice versa.
- Integration of both these system can produce systems that are generalized yet probabilistic in nature.

PROPOSED MODELS

- Probabilistic Relational Model
- Object Oriented Bayesian Network
- Bayesian Program Logic
- Probabilistic Ontology

Relational Probabilistic Models

- PRM/PER has its roots in relational model
- Perceives data in the form of relation ( tabular structure)
- PER model attempted to integrate multiple instance capability of relational model with probabilistic support of Bayesian network
- DAPER is an ER model with directed (solid) arcs among the attribute classes that represent probabilistic dependencies among corresponding attributes, and local distribution classes that define local distributions for attributes.
- DAPER, Plate and PRM are examples of relational probabilistic models

Probabilistic Relational Models

Ref: Probabilistic Entity-Relationship Models, PRMs and Plate Models, David Heckerman, Christopher Meek, and Daphne Koller, 2007

Object Oriented Bayesian Network

- OOBN allows knowledge representation in the form of classes.
- Classes represent complex objects that in turn are composed of other simple/complex objects.
- Set of classes typically forms a is-a or part-of hierarchy in the system.
- Each class contains several properties that are categorized as Input/Hidden/Output properties. Input and Output properties define interface of class.
- Each object of class is considered a stochastic function that transforms particular values of Input attributes to output attributes.

Bayesian Logic Program

- BLP has its roots in First order logic
- Instead of having predicates, Bayesian logic program comprises of Bayesian clauses.
- Bayesian clause can have different states with varying degree of belief.
- Bayesian clause has an associated conditional probability table and each Bayesian predicate has an associated combining rule that maps finite set of multiple probability distributions onto one probability distribution.

Equivalent graphical representation

[Ref: Bayesian logic programming: Theory & Tool KristianKersting and Luc De Raedt]

Bayesian Logic Program

Propositional Representation:

A(s1) Ù CT (c1) Ù Q (t1) ® G (s1, c1)

A(s1) Ù C (c2) Ù Q (t2) ® G (s1, c2)

A(s1) Ù C (c3) Ù Q (t3) ® G (s1, c3)

G (s1,c1) . G (s1,c2). G (s1,c3) ® P (s1)

Predicate-based Representation:

A(X) Ù C (Y) Ù Q (Z) ® G (X,Y)

- r | r Î RC, G(X,c) ® P (X)

A(X) Ù C (Y) Ù Q (Z) ®G (X,Y)

- r | r Î RC, G(X,c) ®P (X)
- v| v Î B, P(v) ® P (B)

[ A = Aptitude, CT = Type of Course, Q = Quality of Teaching, G = Grade, P = Performance,

RC=Registered Courses, B= Batch]

Probabilistic Ontologies

- Ontology is a formal representation of a set of concepts within a domain and the relationship between these concepts. Its is used define particulars of a domain and to reason about the properties of it.
- Probabilistic ontologies expand the possibilities of standard ontologies by introducing the requirement of a proper representation of the statistical regularities and the uncertain evidence about entities in a domain of application and also allowing for reasoning upon what now can be represented via probabilistic ontologies.
- Multi-Entity Bayesian Network (MEBN) has been used a framework for defining probabilistic ontologies and making use of them in inferences.

What is MEBN ?

- MEBN is not a computer language such as Java or C++, or an application such as Netica or Hugin. Rather, it is formal system that combines expressive power of FOL with logically consistent treatment of uncertainty.
- MEBN provides syntax, a set of model construction and inference processes, and semantics that together provide a means of defining probability distributions over unbounded and possibly infinite numbers of interrelated hypotheses

Multi-Entity Bayesian Network

“MEBN is to Bayesian networks as algebra is to arithmetic”

- Each MEBN model, called MTheory, represents a particular domain of discourse.
- Different Subjects of that domain are represented by smaller components known as MFrag. MFrag provides grouping of entities and their relationships pertinent to that particular subject.
- Each node in MFrag is parameterized hence providing support for the multiple instances of it. An MFrag node can be of any of three types:
- Context Node that is evaluated to either true/false when substituted with constant values in place of parameters.
- Resident Nodes are local nodes of MFrag and form the real core of it. There can be multiple resident nodes in an MFrag and each resident node defines its own probability distribution. Semantically, MFrag is a representation of group of its resident nodes. Resident nodes can in turn be dependent on the other instance of their own hence providing support for recursive MFrags.
- Input Nodes serve as input to derive probability distribution of resident nodes. Input nodes are at the interface of local MFrag and are resident nodes of some other MFrag where their own probability distribution is defined. Hence Input nodes provide mechanism to connect multiple MFrags.

Ref: Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA

Figure shows Danger when four ships are nearby

Ref: Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA

This table shows Danger To Self MFrag Probability Distribution

Relevant Starships Nearby Danger Level Dist.

At least 1 Cardassian [0.925, 0.024, 0.006, 0]

At least 2 Cardassians [0.99, 0.008, 0.002, 0]

At least 3 Cardassians [0.975, 0.2, 0.05, 0]

More than 4 Cardassians [1, 0, 0, 0]

No Cardassians but at least 1 Romulan [.73, .162, .081, .027]

No Cardassians but at least 1 Romulans [.76, .144, .072, .024]

… … (see formula)

No Cardassians but 10 or more Romulans [1, 0, 0, 0]

No Cardassians or Romulans, one Unknown [.02, .48, .48, .02]

… … (see formula)

No Cardassians or Romulans, 10+ Unknown [.20, .30, .30, .20]

… …(see formula)

MTHeory Diagram

Ref: Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA

Example - MEBN Query

Finding:

In our example, the finding MFrags will convey information that we have five starships (!ST0 through !ST4) and that the first is our own starship. For the sake of illustration, let’s assume that our Finding set also includes data regarding the nature of the space zone we are in (!Z0), its magnetic disturbance for the first time step (!T0), and sensor reports for starships !SR1 to !SR4 for the first two time steps.

Targets:

We assume that the Target set for our illustrative query includes an assessment of the level of danger experienced by the Enterprise and the best decision to take given this level of danger.

Conclusion

- Full integration of first-order logic with Bayesian Network can enable us to:
- Provide a true representation of domain of discourse that can dynamically generate multiple instances depending upon the situation in hand.
- Capture statistical regularities of that domain
- Make inferences or diagnose causes given certain evidences

References

[1] Bayesian logic programming: Theory & Tool, KristianKersting and Luc De Raedt.

[2] A Dynamic Approach to Probabilistic Inference using Bayesian Networks, Michael C. Horsch and David Poole, Department of Computer Science, University of British Columbia, Canada

[3] First-order probabilistic inference, David Poole in Proceedings IJCAI 2003. Acapulco, Mexico, August 2003, pages 985-991.

[4] Probabilistic Entity-Relationship Models, PRMs and Plate Models, David Heckerman, Christopher Meek, and Daphne Koller, 2007

[5] Bayesian networks and influence diagrams, A guide to Construction and Analysis, Uffe B. Kjærulff • Anders L. Madsen

[6] Koller, D., & Pfeffer, A. (1997). Object-Oriented Bayesian Networks. Paper presented at the Thirteenth Conference on Uncertainty in Artificial Intelligence (UAI-97). San Francisco, CA, USA.

[7] Laskey, K.B., MEBN: A Language for First-Order Bayesian Knowledge Bases, Artificial Intelligence, 172(2-3), 2007.

[8] Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA

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