relational probabilistic graphical modelsesucar/clases-mgp/notes/c12-rpm.pdf · markov logic...

Introduction

ProbabilisticRelationalModelsRepresentation

Inference

Learning

Markov LogicNetworksRepresentation

Inference

Learning

ApplicationsStudent Modeling

Visual Grammars

References

Relational Probabilistic Graphical Models

Probabilistic Graphical Models

L. Enrique Sucar, INAOE

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Outline

1 Introduction

2 Probabilistic Relational ModelsRepresentationInferenceLearning

3 Markov Logic NetworksRepresentationInferenceLearning

4 ApplicationsStudent ModelingVisual Grammars

5 References

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Introduction

Introduction• The standard probabilistic graphical models have to

represent explicitly each object in the domain, so theyare equivalent in terms of their logical expressive powerto propositional logic

• There are problems in which the number of objects(variables) could increase significantly, so a moreexpressive (compact) representation is desirable – i.e.model a student’s knowledge of a certain topic (studentmodeling or, in general, user modeling), and that wewant to include in the model all of the students in acollege, where each student is enrolled in several topics

• It would be more efficient if in some way we could havea general model that represents the dependencyrelations for any student, S and any course, C, whichcould then be parameterized for particular cases

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Introduction

PRGMs

• Relational probabilistic graphical models (RPGMs)combine the expressive power of predicate logic withthe uncertain reasoning capabilities of probabilisticgraphical models

• Some of these models extend PGMs such as Bayesiannetworks or Markov networks by representing objects,their attributes, and their relations with other objects

• Other approaches extend the logic–basedrepresentations, in order to incorporate uncertainty, bydescribing a probability distribution over logic formulas

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Introduction

Types of PRGMs

• Extensions of logic models

1 Undirected graphical models

1 Markov Logic Networks

2 Directed graphical models

1 Bayesian Logic Programs2 Bayesian Logic Networks

• Extensions of probabilistic models

1 Undirected graphical models

1 Relational Markov Networks2 Relational Dependency Networks3 Conditional Random Fields

2 Directed graphical models1 Relational Bayesian Networks2 Probabilistic Relational Models

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Introduction

Type of PRGMs

• Extensions of programming languages

1 Stochastic Logic Programs2 Probabilistic Inductive Logic programming3 Bayesian Logic (BLOG)4 Probabilistic Modeling Language (IBAL)

• In this chapter we will review two of them:• Probabilistic relational models extend Bayesian

networks to incorporate objects and relations, as in arelational data base

• Markov logic networks, which add weights to to logicalformulas, and can be considered as an extension ofMarkov networks

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Probabilistic Relational Models

Probabilistic Relational Models

• Probabilistic relational models (PRMs) are an extensionof Bayesian networks that provide a more expressive,object-oriented representation

• For the case of a very large model, only part of it isconsidered at any time, so the inference complexity isreduced

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Probabilistic Relational Models Representation

Representation

• The basic entities in a PRM are objects or domainentities, whcih are partitioned into a set of disjointclasses X1, ...,Xn

• Each class is associated with a set of attributes A(Xi)

• Each attribute Aij ∈ A(Xi) (that is, attribute j of class i)takes on values in some fixed domain of values V (Aij)

• A set of relations, Rj , are defined between the classes• The classes and relations define the schema of the

model• The dependency model is defined at the class level,

allowing it to be used for any object in the class

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Dependency model

• A PRM specifies the probability distribution using thesame underlying principles used in Bayesian networks

• Each of the random variables in a PRM, the attributesx .a of the individual objects x , is directly influenced byother attributes, which are its parents

• A PRM, therefore, defines for each attribute, a set ofparents, which are the directed influences on it, and alocal probabilistic model that specifies probabilisticparameters

• There are two differences between PRMs and BNs: (i)In a PRM the dependency model is specified at theclass level, allowing it to be used for any object in theclass. (ii) A PRM explicitly uses the relational structureof the model, allowing an attribute of an object todepend on attributes of related objects

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Example - school domain

• There are 4 classes, with 2 attributes each in thisexample:

Professor: teaching-ability, popularityStudent: intelligence, rankingCourse: rating, difficulty

Registration: satisfaction, grade

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Example

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Types of attributes

• This representation allows for two types of attributes ineach class: (i) information variables, (ii) randomvariables.

• The random variables are the ones that are linked in akind of Bayesian network that is called a skeleton

• From this skeleton, different Bayesian networks can begenerated, according to other variables in the model

• This gives the model a greater flexibility and generality,facilitating knowledge acquisition

• It also makes inference more efficient, because onlypart of the model is used in each particular case

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Parameters

• The probability distribution for the skeletons arespecified as in Bayesian networks

• A PRM defines for each attribute x .a, a set of parents,which are the directed influences on it, and a localprobabilistic model that specifies the conditionalprobability of the attribute given its parents

• To guarantee that the local models define a coherentglobal probability distribution, the underlyingdependency structure should be acyclic, as in a BN

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Probabilistic Relational Models Inference

Inference

• The inference mechanism for PRMs is the same as forBayesian networks, once the model is instantiated toparticular objects in the domain

• PRMs can take advantage of two properties to makeinference more efficient

• One property is the locality of influence, most attributeswill tend to depend mostly on attributes of the sameclass, and there are few interclass dependencies – cantake advantage of this locality property by using a divideand conquer approach

• The other aspect is reuse. In a PRM there are usuallyseveral objects of the same class, with similar structureand parameters. Once inference is performed for oneobject, this can be reused for the other similar objects

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Probabilistic Relational Models Learning

Learning

• Given that PRMs share the same underlying principlesof BNs, the learning techniques developed for BNs canbe extended for PRMs

• The expectation maximization algorithm has beenextended to learn the parameters of a PRM, andstructure learning techniques have been developed tolearn the dependency structure from a relationaldatabase

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Markov Logic Networks

Markov Logic Networks

• In contrast to PRMs, Markov logic networks (MLN) startfrom a logic representation, adding weights to formulasto incorporate uncertainty

• In logic, a L-interpretation which violates a formulagiven in a knowledge base (KB) has zero probability

• In Markov Logic Networks, this assumption is relaxed. Ifthe interpretation violates the KB, it has less probabilitythan others with no violations

• In a MLN, a weight to each formula is added in order toreflect how strong the constraint is

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Markov Logic Networks Representation

Markov Networks - brief review

• A Markov Network is a model for the joint distribution ofa set of variables X = (X1,X2, ...,Xn) ∈ X

• It is composed of an undirected graph G with a nodeper variable, and a set of potential functions φk , one foreach clique

• The joint distribution represented by a Markov networkis given by

P(X = x) =1z

∏k

φk (x{k}) (1)

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Markov networks

• Markov Networks can also be represented usinglog-linear models, where each clique potential functionis replaced by an exponential weighted sum:

P(X = x) =1z

exp∑

j

wj fj(x) (2)

• Where wj is a weight (real value) and fj is, for ourpurposes, a binary formula fj(x) ∈ {0,1}

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MLN

• An MLN L is a set of pairs (Fi ,wi), where Fi is a formulain first-order logic and wi is a real number

• Together with a finite set of constantsC = {c1, c2, ..., c|C|}, it defines a Markov network ML,C :

1 ML,C contains one binary node for each possiblegrounding of each formula appearing in the MLN L. Thevalue of the node is 1 if the ground atom is true, and 0otherwise.

2 ML,C contains one feature for each possible grounding ofeach formula Fi in L. The value of this feature is 1 if theground formula is true, and 0 otherwise. The weight ofthe feature is the wi associated with Fi in L.

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Ground MLN

• MLNs are a generalization of Markov networks, so theycan be seen as templates for constructing Markovnetworks

• Given a MLN and a set of different constants, differentMarkov networks can be produced; these are known asground Markov networks

• The joint probability distribution of a ground Markovnetwork is defined in a similar way as a Markov network

• The graphical structure of a MLN is based on itsdefinition; there is an edge between two nodes of theMLN if the corresponding ground atoms appeartogether in at least one grounding of one formula in theknowledge base

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MLN - example

• MLN consisting of two logical formulas:

∀xSmoking(x)→ Cancer(x)

∀x∀yFriends(x)→ (Smoking(x)↔ Smoking(y))

• x and y are instantiated to the constants A and B

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Markov Logic Networks Inference

Inference

• Inference in MLN consists in estimating the probabilityof a logical formula, F1, given that another formula (orformulas), F2, are true

• That is, calculating P(F1 | F2,L,C), where L is a MLNconsisting of a set of weighted logical formulas, and Cis a set of constants

• To compute this probability, we can estimate theproportion of possible worlds in which F1 and F2 aretrue, over the possible worlds in which F2 holds – theprobability of each possible world is consideredaccording to the weights of the formulas and thestructure of the grounded Markov network

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Markov Logic Networks Inference

Inference

• Performing the previous calculations directly is,computationally, very costly

• One alternative is using stochastic simulation; bysampling the possible worlds we can obtain an estimateof the desired probability; for instance, using the Markovchain Montecarlo techniques

• Another alternative is to make certain reasonableassumptions about the structure of the logical formulasthat simplify the inference process – for instance that F1and F2 are conjunctions of ground literals

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Markov Logic Networks Learning

Learning

• Learning a MLN involves two aspects. One aspect islearning the logical formulas (structure), and the other islearning the weights for each formula (parameters)

• For learning the logical formulas, we can applytechniques from the area of inductive logicprogramming (ILP). There are different approaches thatcan induce logical relations from data, consideringsome background knowledge

• The weights of the logical formulas can be learned froma relational database, the weight of a formula isproportional to its number of true groundings in the datawith respect to its expectation according to the model(can also use sampling)

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Applications

Applications

• A general student model for virtual laboratories basedon PRMs

• MLNs for representing visual grammars for objectrecognition

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Applications Student Modeling

Student Modeling with PRMs

• A virtual lab provides a simulated model of someequipment, so that students can interact with it andlearn by doing

• A tutor serves as a virtual assistant in this lab, providinghelp and advice to the user, and setting the difficulty ofthe experiments, according to the student’s level

• The cognitive state should be obtained based solely onthe student’s interactions with the virtual lab and theresults of the experiments – a student model

• The model infers, from the student’s interactions withthe laboratory, the cognitive state; and based on this, anintelligent tutor can give personalized advice to thestudent

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Probabilistic relational student model

• PRMs provide a compact and natural representation forstudent modeling

• Each class represents the set of parameters of severalstudents, like in databases, but the model also includesthe probabilistic dependencies between classes foreach student

• In order to apply PRMs to student modeling we have todefine the main objects involved in the domain

• The dependency model is defined at the class level,allowing it to be used for any object in the class

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High level model

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ClassesStudent: student-id, student-name, major, quarter,

category.Knowledge Theme: student-id, knowledge-theme-id,

knowledge-theme-known.Knowledge Sub-theme: student-id,

knowledge-sub-theme-id,knowledge-sub-theme-known.

Knowledge Items: student-id, knowledge-item-id,knowledge-item-known.

Academic background: previous-course, grade.Student behavior: student-id, experiment-id, behavior-var1,

...Experiment results: student-id, experiment-id,

experiment-repetition, result-var1, ...Experiments: experiment-id, experiment-description,

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Skeleton

• From the PRM student model we can define a generalBayesian network, a skeleton, that can be instantiatedfor different scenarios, in this case experiments

• From the class model we obtain a hierarchical skeleton• From the skeleton, it is possible to define different

instances according to the values of specific variables inthe model

• We can define particular instances for each experiment(for example, in the robotics domain, there could beexperiments related to robot design, control, motionplanning, etc.) and student level (novice, intermediate,advanced)

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Skeleton

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Inference

• Once a specific Bayesian network is generated, it canbe used to update the student model via standardprobability propagation techniques

• In this case, it is used to propagate evidence from theexperiment evaluation to the knowledge items, and tothe knowledge sub-themes and to the knowledgethemes

• This is used by the tutor to decide if it should providehelp to the student, and at what level of detail

• For example, if in general the experiment wassuccessful, but some aspect was not very good, alesson on a specific concept (item) is given to thestudent

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Applications Visual Grammars

Visual Grammars

• A visual grammar describes objects hierarchically• For visual object recognition, we need a grammar that

allows us to model the decomposition of a visual objectinto its parts and how they relate with each other

• One interesting kind of relational grammar areSymbol-Relation Grammars

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Object representation

• Classes of objects are represented based onsymbol-relational grammars

• This includes three basic parts: (i) the basic elements ofthe grammar or lexicon, (ii) the spatial relations, (iii) thetransformation rules

• The visual features considered are: uniform colorregions (color is quantized in 32 levels) and edges atdifferent orientations (obtained with Gabbor filters) –these are clustered and the centroids of these clustersconstitute the Visual Lexicon

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Relations

• The spatial relations include topological and orderrelationships

• The relationships used are: Inside of (A,B) (A region iswithin B region), Contains(A,B) (A region coverscompletely B region), Left(A,B) (A is touched by B andA is located left from B), Above(A,B) (A is touched by Band A is located above from B), Invading(A,B) (A ispartially covering B more than Above and Left but lessthan Contains)

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Rules

• The next step is to generate the rules that make up thegrammar

• Using training images for the class of the object ofinterest, the most common relationships betweenclusters are obtained – become candidate rules to buildthe grammar

• This is an iterative process where the rules aresubsumed and converted to new non-terminal elementsof the grammar

• The starting symbol of the grammar represents theclass of objects to be recognized

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Transforming a SR Grammar into a MarkovLogic Network

• The SR grammar for a class of objects is transformeddirectly to formulas in the MLN language – structure ofthe model

• The parameters –weights associated to each formula–,are obtained from the training image set.

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Example - MLN to recognize faces

• SR grammar to recognize faces based on high-level features:eyes, mouth, nose, head

• The productions of this simple SR-grammar for faces are:

1 : FACE0 → < {eyes2,mouth2}, {above(eyes2,mouth2)} >2 : FACE0 → < {nose2,mouth2}, {above(nose2,mouth2)} >3 : FACE0 → < {eyes2,head2}, {inside of (eyes2,head2)} >4 : FACE0 → < {nose2,head2}, {inside of (nose2,head2)} >5 : FACE0 → < {mouth2,head2}, {inside of (mouth2,head2)} >

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Transformation to MLN

First, we need to declare the formulas:aboveEM(eyes,mouth)

aboveNM(nose,mouth)

insideOfEH(eyes,head)

insideOfNH(nose,head)

insideOfMH(mouth,head)

isFaceENMH(eyes,nose,mouth,head)

Subsequently, we need to declare the domain:eyes={E1,E2,E3,E4}nose={N1,N2,N3,N4}mouth={M1,M2,M3,M4}head={H1,H2,H3,H4}

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Transformation to MLN

Finally we need to write the weighted first-order formulas.We used a validation image dataset and translated theprobabilities into weights:1.58 isFaceENMH(e,n,m,h) => aboveEM(e,m)

1.67 isFaceENMH(e,n,m,h) => aboveNM(n,m)

1.16 isFaceENMH(e,n,m,h) => insideOfEH(e,h)

1.25 isFaceENMH(e,n,m,h) => insideOfNH(n,h)

1.34 isFaceENMH(e,n,m,h) => insideOfMH(m,h)

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Recognition

• To recognize a face in an image, the relevant aspects ofthe image are transformed into a first-order KB

• For this, the terminal elements are detected (in theexample the eyes, mouth, nose and head) in the image,as well as the spatial relations between these elements

• Then, the particular image KB is combined with thegeneral model represented as a MLN; and from thiscombination a grounded Markov network is generated

• Object (face) recognition is performed using standardprobabilistic inference over the Markov network

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Book

Sucar, L. E, Probabilistic Graphical Models, Springer 2015 –Chapter 12

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Additional Reading (1)

Domingos, P., Richardson, M.: Markov Logic: A UnifyingFramework for Statistical Relational Learning. In: L.Getoor, B. Taskar (eds.) Introduction to StatisticalRelational Learning, pp. 339–371. MIT Press,Cambridge (2007)

Ferrucci, F., Pacini, G., Satta, G., Sessa, M. I., Tortora,G., Tucci, M., Vitiello, G.: Symbol-Relation Grammars: AFormalism for Graphical Languages. Information andComputation, 131(1), 1-46 (1996)

Friedman, N., Getoor, L., Koller, D., Pfeffe, A.: LearningProbabilistic Relational Models. In: Proceeding of theInternational Joint Conference on Artificial Intelligence(IJCAI), pp.1300–1309 (1999)

Genesereth, M.R., Nilsson, N.J.: Logical Foundations ofArtificial Intelligence. Morgan Kaufmann (1988)

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Getoor, L., Taskar, B.: Introduction to StatisticalRelational Learning. MIT Press, Cambridge (2007)

Koller D.: Probabilistic Relational Models. In:Proceedings of the 9th International Workshop onInductive Logic Programming, Lecture Notes in ArtificialIntelligence 1634, Springer, 3–13 1999

Lavrac, N., Dzeroski, S.: Inductive Logic Programming:Techniques and Applications. Ellis Horwood, New York(1994)

Lemmon, E.J.: Beginning Logic. Hackett PublishingCompany (1978)

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Richardson, M., Domingos, P.: Markov Logic Networks.Machine Learning. 62(1-2), 107–136 (2006)

Ruiz, E., Sucar, L.E.: An Object Recognition ModelBased on Visual Grammars and Bayesian Networks. In:Proceedings of the Pacific Rim Symposium on Imageand Video Technology, LNCS 8333, pp. 349–359,Springer-Verlag (2014)

Nienhuys-Cheng, S., de Wolf, R.: Foundations ofInductive Logic Programming. Springer-Verlag, Berlin(1991)

Sucar, L.E., Noguez, J.: Student Modeling. In: O.Pourret, P. Naim, B. Marcot (eds.) Bayesian BeliefNetworks: A Practical Guide to Applications, pp.173–186. Wiley and Sons (2008)

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