Presentation on theme: "First Order Bayesian Network"— Presentation transcript:
1First Order Bayesian Network Literature ReviewPresented By:Syeda Saleha RazaFaculty of Computer ScienceIBA, Karachi
2Reasoning 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
8Limitations of Bayesian Networs Ability to create multiple instances of same nodeEffective mechanism to specify frequency distribution for a node having unbounded number of parentsAbility to quantify (existentially & universally) over unbounded and potentially infinite number of parents for a given nodeAbility to consider size of population that is not part of domain but is known to exist
9Proposition 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)
10Limitations of FOLSystems based on FOL have the ability to represent entities of different types interacting with each other in varied waysHowever systems based on FOL lack theoretically principled, widely accepted, logically coherent methodology for reasoning under uncertainty
11Bayesian 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 natureThe 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.
12PROPOSED MODELS Probabilistic Relational Model Object Oriented Bayesian NetworkBayesian Program LogicProbabilistic Ontology
13Relational Probabilistic Models PRM/PER has its roots in relational modelPerceives data in the form of relation ( tabular structure)PER model attempted to integrate multiple instance capability of relational model with probabilistic support of Bayesian networkDAPER 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
14Probabilistic Relational Models Ref: Probabilistic Entity-Relationship Models, PRMs and Plate Models, David Heckerman, Christopher Meek, and Daphne Koller, 2007
15Object 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.
16Bayesian Logic Program BLP has its roots in First order logicInstead 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.
17Equivalent graphical representation [Ref: Bayesian logic programming: Theory & Tool Kristian Kersting and Luc De Raedt]
18Bayesian 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)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]
19Probabilistic 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.
20What 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
21Multi-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.
22Ref: Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web Ref: Costa, P. C. G Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA
23Figure shows Danger when four ships are nearby Ref: Costa, P. C. G Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA
24This 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]No Cardassians or Romulans, 10+ Unknown [.20, .30, .30, .20]… …(see formula)
25MTHeory DiagramRef: Costa, P. C. G Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA
26Example - MEBN QueryFinding: 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.
27Ref: Costa, P. C. G. 2005. Bayesian Semantics for the Semantic Web Ref: Costa, P. C. G Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA
28ConclusionFull 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 domainMake inferences or diagnose causes given certain evidences
29References Bayesian logic programming: Theory & Tool, Kristian Kersting and Luc De Raedt. A Dynamic Approach to Probabilistic Inference using Bayesian Networks, Michael C. Horsch and David Poole, Department of Computer Science, University of British Columbia, Canada First-order probabilistic inference, David Poole in Proceedings IJCAI Acapulco, Mexico, August 2003, pages Probabilistic Entity-Relationship Models, PRMs and Plate Models, David Heckerman, Christopher Meek, and Daphne Koller, 2007 Bayesian networks and influence diagrams, A guide to Construction and Analysis, Uffe B. Kjærulff • Anders L. Madsen 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. Laskey, K.B., MEBN: A Language for First-Order Bayesian Knowledge Bases, Artificial Intelligence, 172(2-3), 2007. Costa, P. C. G Bayesian Semantics for the Semantic Web. PhD Diss. Department of Systems Engineering and Operations Research, George Mason University. 315p, July 2005, Fairfax, VA, USA