Markov Logic Networks

Speaker:Benedict FehringerSeminar:Probabilistic Models for Information Extractionby Dr. Martin Theobald and Maximilian Dylla

Based on Richards, M., and Domingos, P. (2006)Markov Logic Networks1Markov Logic NetworksOutlineMarkov Logic Networks2 Part 1: Why we need Markov Logic Networks (MLNs) Markov Networks First-Order Logic Conclusion and Motivation Part 2: How the MLNs work? Part 3: Are they better than other methods?

OutlineMarkov Logic Networks3 Part 1: Why we need Markov Logic Networks (MLNs) Markov Networks First-Order Logic Conclusion and Motivation Part 2: How the MLNs work? Part 3: Are they better than other methods?

Markov NetworksMarkov Logic Networks4Set of variables:

The distribution is given by:

with as normalization factorand as potential function

Markov NetworksMarkov Logic Networks5Representation as log-linear model:

In our case there will be only binary features:Each feature corresponds to each possible state

The weight is equal to the log of the potential:

Little example scenarioMarkov Logic Networks6There is a company witch has a Playstation and each companys employee has the right to play with it.If two employees are friends then the probability is high (=3) that both play all day long or both do not.If someone plays all day long then the chance for her/him is high (=2) to get fired.Little example scenarioMarkov Logic Networks7There is a company witch has a Playstation and each companys employee has the right to play with it.If two employees are friends then the probability is high (=3) that both play all day long or both do not.If someone plays all day long then the chance for her/him is high (=2) to get fired.Markov NetworksMarkov Logic Networks8Has playing FriendPlaysFiredPlaysFiredTrueTrue2TrueFalse0FalseTrue2FalseFalse2FriendsPlaysTrueTrue3TrueFalse0FalseTrue0FalseFalse3One possibility

Markov NetworksMarkov Logic Networks9Is Friend withPlaysFiredSome playing EmployeeAnd another one.Little example scenarioMarkov Logic Networks10There is a company witch has a Playstation and each companys employee has the right to play with it.If two employees are friends then the probability is high (=3) that both play all day long or both do not.If someone plays all day long then the chance for her/him is high (=2) to get fired.

If an employee A can convince another employee B to play, depends on the lability of B. For a high lability of B there is a higher probability (=4) than for a low lability (=2)Markov NetworksMarkov Logic Networks11Is Friend withPlaysFiredPlaying PersonIs labileCould be.Markov NetworksMarkov Logic Networks12Advantages: Efficiently handling uncertainty Tolerant against imperfection and contradictory knowledge

Disadvantages: Very complex networks for a wide variety of knowledge Difficult to incorporate a wide range of domain knowledge

OutlineMarkov Logic Networks13 Part 1: Why we need Markov Logic Networks (MLNs) Markov Networks First-Order Logic Conclusion and Motivation Part 2: How the MLNs work? Part 3: Are they better than other methods?

First-Order LogicMarkov Logic Networks14Four types of symbols:Constants:concrete object in the domain (e.g., people: Anna, Bob)Variables:range over the objects in the domainFunctions:Mapping from tuples of objects to objects (e.g., GrandpaOf)Predicates:relations among objects in the domain (e.g., Friends) or attributes of objects (e.g. Fired)Term: Any expression representing an object in the domain. Consisting of a constant, a variable, or a function applied to a tuple of terms.Atomic formula or atom: predicate applied to a tuple of termsLogical connectives and quantifier:

Translation in First-Order LogicMarkov Logic Networks15If two employees are friends then the probability is high that both play all day long or both do not.

In Clausal Form:

If someone plays all day long then the chance for her/him is high to get fired.

In Clausal Form:

First-Order LogicMarkov Logic Networks16Advantages: Compact representation a wide variety of knowledge Flexible and modularly incorporate a wide range of domainknowledge

Disadvantages: No possibility to handle uncertainty No handling of imperfection and contradictory knowledge

OutlineMarkov Logic Networks17 Part 1: Why we need Markov Logic Networks (MLNs) Markov Networks First-Order Logic Conclusion and Motivation Part 2: How the MLNs work? Part 3: Are they better than other methods?

Conclusion and MotivationMarkov NetworksFirst-Order LogicMarkov Logic Networks18Efficiently handling uncertaintyTolerant against imperfection and contradictory knowledge

Compact representation a wide variety of knowledgeFlexible and modularly incorporate a wide range of domain knowledge Combination of Markov Networks and First-Order Logic to use the advantages of both 18OutlineMarkov Logic Networks19 Part 1: Why we need Markov Logic Networks (MLNs) Markov Networks First-Order Logic Conclusion and Motivation Part 2: How the MLNs work? Part 3: Are they better than other methods?

Markov Logic NetworkMarkov Logic Networks20Description of the problemTranslation in First-Order LogicConstruction of a MLN-TemplateDerive a concrete MLN for a given Set of ConstantsCompute whatever you wantMarkov Logic NetworkMarkov Logic Networks21Description of the problemTranslation in First-Order LogicConstruction of a MLN-TemplateDerive a concrete MLN for a given Set of ConstantsCompute whatever you wantMarkov Logic Network- Translation in First-Order Logic -Markov Logic Networks22If two employees are friends then the probability is high that both play all day long or both do not.

In Clausal Form:

If someone plays all day long then the chance for her/him is high to get fired.

In Clausal Form:

Markov Logic NetworkMarkov Logic Networks23Description of the problemTranslation in First-Order LogicConstruction of a MLN-TemplateDerive a concrete MLN for a given Set of ConstantsCompute whatever you wantMarkov Logic NetworkMarkov Logic Networks24Each formula matches one cliqueEach formula owns a weight that reflects the importance of this formulaIf a world violates one formula then it is less probable but not impossibleConcrete: The weight of this formula will be ignored (that means the weight is 0)

Markov Logic NetworkMarkov Logic Networks25Formula

to compare:

Three Assumptions:Unique NamesDomain closureKnown functions

Markov Logic NetworkMarkov Logic Networks26Description of the problemTranslation in First-Order LogicConstruction of a MLN-TemplateDerive a concrete MLN for a given Set of ConstantsCompute whatever you wantMarkov Logic NetworkMarkov Logic Networks27Grounding: (with Constants c1 and c2)

=>

=>

=>Elimination of the existential quantifierElimination of the universal quantifierElimination of the functions

Plays(c1)Plays(c2)

Fired(c1)Fired(c2)

TrueFalseTrueFalse

Markov NetworksMarkov Logic Networks28Friends(A,A)Friends(B,B)Friends(A,B)Friends(B,A)Plays(A)Plays(B)

Constants: Alice (A) and Bob (B)Markov Logic NetworkMarkov Logic Networks29Friends(A,A)Friends(B,B)Friends(A,B)Friends(B,A)Plays(A)Plays(B)Fired(B)Fired(A)

Constants: Alice (A) and Bob (B)Plays(A)Plays(B)Markov Logic NetworkMarkov Logic Networks30Friends(A,A)Friends(B,B)Friends(A,B)Friends(B,A)Plays(A)Plays(B)Fired(B)Fired(A)

Friends(x,y)Plays(x)Plays(y)

TrueTrueTrue3TrueFalseTrue0FalseTrueTrue3FalseFalseTrue3TrueTrueFalse0TrueFalseFalse3FalseTrueFalse3FalseFalseFalse3Plays(x)Fired(x)TrueTrue2TrueFalse0FalseTrue2FalseFalse2Markov Logic NetworkMarkov Logic Networks31Description of the problemTranslation in First-Order LogicConstruction of a MLN-TemplateDerive a concrete MLN for a given Set of ConstantsCompute whatever you wantMarkov Logic NetworkMarkov Logic Networks32What is the probability that Alice and Bob are friends, both play playstation all day long but both are not getting fired?

Friends(A,A)=1Friends(B,B)=1Friends(A,B)=1Friends(B,A)=1Plays(A)=1Plays(B)=1Fired(B)=0Fired(A)=0Friends(A,B)Plays(A)Plays(B)

TrueTrueTrue3Plays(x)Fired(x)TrueFalse0

Friends(x,x)Plays(x)FalseTrue3Markov Logic NetworkMarkov Logic Networks33What happens if lim?

If all formulas fulfilled:

If not all formulas fulfilled:

=>

Markov Logic NetworkMarkov Logic Networks34What happens if lim?

If all formulas fulfilled:

Markov Logic NetworkMarkov Logic Networks35What happens if lim?

If not all formulas fulfilled:

Markov Logic NetworkMarkov Logic Networks36What is the probability that a formula F1 holds given that formula F2 does?

Markov Logic NetworkMarkov Logic Networks37Learning the weights:It is #P-complete to count the number of true groundings=> approximation is necessary=> using the pseudo-likelihood:

OutlineMarkov Logic Networks38 Part 1: Why we need Markov Logic Networks (MLNs) Markov Networks First-Order Logic Conclusion and Motivation Part 2: How the MLNs work? Part 3: Are they better than other methods?

Experiment IMarkov Logic Networks39Setting:Using a database describing the Department of Computer Science and Engineering at the University of Washington12 predicates (e.g. Professor, Student, Area, AdvisedBy, )2707 constants96 formulas (Knowledge base was provided by four volunteers who not know the database but were member of the department)The whole database was divided into five subsets for each area (AI, graphics, programming languages, systems, theory)=> in the end 521 true ground atoms of possible 58,457

Experiment IIMarkov Logic Networks40Testing:leave-one-out over the areasPrediction of AdvisedBy(x,y)Either with all or only partial (except Student(x) and Professor(x)) informationDrawing the precision/recall curvesComputation of the area under the curve (AUC)Experiment IIIMarkov Logic Networks41MLN was compared with:Logic (only logical KB without probability)Probability (only probability relations without special knowledge representations)Nave Bayes (NB) and Bayesian Network (BN)Inductive logic programming (automatically development of the KB)CLAUDIEN (clausal discovery engine)

ResultsMarkov Logic Networks42

Results IMarkov Logic Networks43

all Areas

AI AreaResults IIMarkov Logic Networks44

graphics areaprog. Language Area

Results IIIMarkov Logic Networks45systems areatheory Area

Sample applicationsMarkov Logic Networks46Link PredictionLink-Based ClusteringSocial Network ModelingConclusionMarkov Logic Networks47MLNs are a simple way to combine first-order logic and probabilityThey can be seen as a template for construction ordinary Markov NetworksClauses can be learned by CLAUDIENEmpirical tests with real-world data and knowledge are promising for the use of the MLNsLiteratureMarkov Logic Networks48Richardson, M., & Domingos, P. (2006). Markov logic networks. Machine Learning Journal, 62, 107-136.