Statistical Relational Artificial Intelligence. Logic, Probability and Computation

163 Pages·2016·1.98 MB·English

by  Luc De Raedt

Statistical Relational
Artificial Intelligence
Logic,Probability,andComputation
Luc De Raedt
KULeuven,Belgium
Kristian Kersting
TechnicalUniversityofDortmund,Germany
Sriraam Natarajan
IndianaUniversity
David Poole
UniversityofBritishColumbia
SYNTHESISLECTURESONARTIFICIALINTELLIGENCE
ANDMACHINE LEARNING#32
M
&C Morgan &cLaypool publishers
Copyright©2016byMorgan&Claypool
StatisticalRelationalArtificialIntelligence:Logic,Probability,andComputation
LucDeRaedt,KristianKersting,SriraamNatarajan,andDavidPoole
www.morganclaypool.com
ISBN:9781627058414 paperback
ISBN:9781627058421 ebook
DOI10.2200/S00692ED1V01Y201601AIM032
APublicationintheMorgan&ClaypoolPublishersseries
SYNTHESISLECTURESONARTIFICIALINTELLIGENCEANDMACHINELEARNING
Lecture#32
SeriesEditors:RonaldJ.Brachman,Yahoo!Labs
WilliamW.Cohen,CarnegieMellonUniversity
PeterStone,UniversityofTexasatAustin
SeriesISSN
Print1939-4608 Electronic1939-4616
ABSTRACT
An intelligent agent interacting with the real world will encounter individual people, courses,
testresults,drugsprescriptions,chairs,boxes,etc.,andneedstoreasonaboutpropertiesofthese
individualsandrelationsamongthemaswellascopewithuncertainty.
Uncertaintyhasbeenstudiedinprobabilitytheoryandgraphicalmodels,andrelationshave
beenstudiedinlogic,inparticularinthepredicatecalculusanditsextensions.isbookexamines
the foundations of combining logic and probability into what are called relational probabilistic
models. It introduces representations, inference, and learning techniques for probability, logic,
andtheircombinations.
e book focuses on two representations in detail: Markov logic networks, a relational
extensionofundirectedgraphicalmodelsandweightedfirst-orderpredicatecalculusformula,and
Problog,aprobabilisticextensionoflogicprogramsthatcanalsobeviewedasaTuring-complete
relationalextensionofBayesiannetworks.
KEYWORDS
probabilisticlogicmodels,relationalprobabilisticmodels,liftedinference,statistical
relational learning, probabilistic programming, inductive logic programming, logic
programming,machinelearning,Prolog,Problog,Markovlogicnetworks
Contents
Preface ...........................................................xiii
1 Motivation......................................................... 1
1.1 UncertaintyinComplexWorlds ......................................1
1.2 ChallengesofUnderstandingStarAI ..................................3
1.3 eBenefitsofMasteringStarAI .....................................5
1.4 ApplicationsofStarAI ..............................................5
1.5 BriefHistoricalOverview ..........................................12
PARTI Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 StatisticalandRelationalAIRepresentations ........................... 17
2.1 ProbabilisticGraphicalModels ......................................17
2.1.1 BayesianNetworks ..........................................18
2.1.2 MarkovNetworksandFactorGraphs ...........................20
2.2 First-OrderLogicandLogicProgramming ............................22
3 RelationalProbabilisticRepresentations ............................... 27
3.1 AGeneralView:ParameterizedProbabilisticModels ....................28
3.2 TwoExampleRepresentations:MarkovLogicAndProbLog ..............34
3.2.1 UndirectedRelationalModel:MarkovLogic .....................35
3.2.2 DirectedRelationalModels:ProbLog ...........................37
4 RepresentationalIssues ............................................. 45
4.1 KnowledgeRepresentationFormalisms................................45
4.2 ObjectivesforRepresentationLanguage ...............................46
4.3 Directedvs.Undirectedmodels......................................48
4.4 First-OrderLogicvs.LogicPrograms.................................50
4.5 FactorsandFormulae..............................................51
4.6 ParameterizingAtoms .............................................52
4.7 AggregatorsandCombiningRules ...................................54
4.8 OpenUniverseModels.............................................58
4.8.1 IdentityUncertainty .........................................58
4.8.2 ExistenceUncertainty ........................................60
4.8.3 Ontologies.................................................62
PARTII Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 InferenceinPropositionalModels .................................... 65
5.1 ProbabilisticInference .............................................65
5.1.1 VariableElimination .........................................66
5.1.2 RecursiveConditioning ......................................66
5.1.3 BeliefPropagation...........................................68
5.2 LogicalInference .................................................69
5.2.1 PropositionalLogic,Satisfiability,andWeightedModelCounting ....69
5.2.2 SemiringInference ..........................................70
5.2.3 eLeastHerbrandModel ...................................72
5.2.4 Grounding.................................................74
5.2.5 Proving ...................................................75
6 InferenceinRelationalProbabilisticModels ............................ 77
6.1 GroundedInferenceforRelationalProbabilisticModels ..................77
6.1.1 WeightedModelCounting....................................77
6.1.2 WMCforMarkovLogic .....................................77
6.1.3 WMCforProbLog..........................................78
6.1.4 KnowledgeCompilation......................................79
6.2 LiftedInference:ExploitingSymmetries ..............................80
6.2.1 ExactLiftedInference .......................................83
6.3 (Lifted)ApproximateInference......................................87
PARTIII Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7 LearningProbabilisticandLogicalModels ............................. 93
7.1 LearningProbabilisticModels.......................................93
7.1.1 FullyObservedDataandKnownStructure .......................94
7.1.2 PartiallyObservedDatawithKnownStructure....................95
7.1.3 UnknownStructureandParameters.............................95
7.2 LogicalandRelationalLearning .....................................96
7.2.1 TwoLearningSettings .......................................97
7.2.2 eSearchSpace............................................98
7.2.3 TwoAlgorithms:ClausalDiscoveryandFOIL ....................99
7.2.4 FromPropositionaltoFirst-OrderLogic........................101
7.2.5 AnILPExample...........................................102
8 LearningProbabilisticRelationalModels ............................. 105
8.1 LearningasInference.............................................105
8.2 eLearningProblem ............................................106
8.2.1 eDataUsed.............................................106
8.3 ParameterLearningofRelationalModels.............................108
8.3.1 FullyObservableData ......................................108
8.3.2 PartiallyObservedData .....................................109
8.3.3 LearningwithLatentVariables ...............................111
8.4 StructureLearningofProbabilisticRelationalModels...................112
8.4.1 AVanillaStructureLearningApproach.........................113
8.4.2 ProbabilisticRelationalModels ...............................114
8.4.3 Boosting .................................................117
8.5 BayesianLearning ...............................................119
PARTIV BeyondProbabilities . . . . . . . . . . . . . . . . . . . . . 121
9 BeyondBasicProbabilisticInferenceandLearning...................... 123
9.1 LiftedSatisfiability...............................................123
9.2 ActinginNoisyRelationalWorlds ..................................124
9.3 RelationalOptimization...........................................129
10 Conclusions...................................................... 135
Preface
is book aims to provide an introduction that can help newcomers to the field to get started,
to understand the state-of-the-art and the current challenges and be ready for future advances.
It reviews the foundations of StarAI, motivates the issues, justifies somechoices that have been
made,andprovidessomeopenproblems.Layingbarethefoundationswillhopefullyinspireothers
tojoinusinexploringthefrontiersandtheyetunexploredareas.
etargetaudienceforthisbookconsistsofadvancedundergraduateandgraduatestudents
andalsoresearchersandpractitionerswhowanttogetanoverviewofthebasicsandthestate-of-
the-artinStarAI.Tothisaim,PartIstartswithprovidingthenecessarybackgroundinprobability
andlogic.Wethendiscusstherepresentationsofrelationalprobabilitymodelsandtheunderlying
issues.Afterward,wefocusfirstoninference,inPartII,andthenonlearning,inPartIII.Finally,
wetouchuponrelationaltasksthatgobeyondthebasicprobabilisticinferenceandlearningtasks
aswellassomeopenissues.
ResearcherswhoarealreadyworkingonStarAI—weapologizetoanyonewhoseworkwe
areaccidentallynotciting—mayenjoyreadingaboutpartsofStarAItheyarelessfamiliarwith.
Wearegratefultoallthepeoplewhocontributedtothedevelopmentofstatisticalrelational
learningandstatisticalrelationalAI.isbookismadepossiblebyyou.
Wealsothankthereviewersfortheirvaluablefeedbackandourco-authors,whoaccompa-
nied us on our StarAI adventures, such as Laura Antanas, Udi Apsel, Babak Ahmadi, Hendrik
Blockeel, Wolfram Burgard, Maurice Bruynooghe, David Buchman, Hung H. Bui, Peter Car-
bonetto, Alexandru Cocora, Fabrizio Costa, Michael Chiang, Walter Daelemans, Jesse Davis,
Nando de Freitas, Kurt De Grave, Tinne De Laet, Bart Demoen, Kurt Driessens, Saso Dze-
roski,omasG.Dietterich,AdamEdwards,AlanFern,DaanFierens,PaoloFrasconi,Roman
Garnett,AmirGloberson,BerndGutmann,MartinGrohe,FabianHadiji,McEloryHoffmann,
ManfredJaeger,GerdaJanssens,orstenJoachims,SaketJoshi,LeslieKaelbling,AndreasKar-
wath,ArzooKatiyar,SeyedM.Kazemi,AngelikaKimmig,JacekKisynski,TusharKhot,Stefan
Kramer,GautamKunapuli,Chia-LiKuo,TobiasLang,NielsLandwehr,DanielLowd,Cather-
ineA.McCarty,eofrastosMantadelis,WannesMeert,BrianMilch,MartinMladenov,Bog-
danMoldovan,RoserMorante,PlinioMoreno,MarionNeumann,DavideNitti,PhillipOdom,
JoseOramas,DavidPage,AndreaPasserini,RuiPimenteldeFigueiredo,ChristianPlagemann,
TapaniRaiko,ChristopherRe,KateRevoredo,AchimRettinger,RicardoRocha,ScottSanner,
Vitor Santos Costa, Jose Santos-Victor, Erkal Selman, Rita Sharma, Jude W. Shavlik, Prasad
Tadepalli,NimaTaghipour,Ingoon,HannuToivonen,PavelTokmakov,SunnaTorge,Marc
Toussaint, Volker Tresp, Tinne Tuytelaars, Vincent Van Asch, Guy Van den Broeck, Martijn
van Otterlo, Joost Vennekens, Jonas Vlasselaer, Zhao Xu, Shuo Yang, and Luke Zettlemoyer.
anksforalltheencouragementandfun!ankstotheStaRAIlabatIndianaforproofreading
thebook.
Last but not least, we also thank our families and friends for their patience and support.
anks!
LDR and KK thank the European Commission for support of the project FP7-248258-
First-MM. KK further thanks Fraunhofer Society, ATTRACT Fellowship ”STREAM”, the
GermanScienceFoundation,DFGKE1686/2-1,aspartoftheDFGPriorityProgramme1527,
andtheGerman-IsraeliFoundationforScientificResearchandDevelopment,GIF1180/2011.
SNthanksArmyResearchOffice(ARO)grantnumberW911NF-13-1-0432undertheYoung
InvestigatorProgramandtheNationalScienceFoundationgrantno.IIS-1343940.LDRthanks
theResearchFoundationFlanders,andtheKULeuvenBOFfundfortheirsupport.DPthanks
theNaturalSciencesandEngineeringResearchCouncilofCanada(NSERC)forongoingsup-
port.
LucDeRaedt,Leuven,Belgium
KristianKersting,Dortmund,Germany
SriraamNatarajan,Bloomington,USA
DavidPoole,Vancouver,Canada
February2016
C H A P T E R 1
Motivation
ere are good arguments that an intelligent agent that makes decisions about how to act in a
complexworldneedstomodelitsuncertainty;itcannotjustactpretendingthatitknowswhatis
true.Anagentalsoneedstoreasonaboutindividuals(objects,entities,things)andaboutrelations
amongtheindividuals.
eseaspectshaveoftenbeenstudiedseparately,withmodelsforuncertaintyoftendefined
in terms of features and random variables, ignoring relational structure, and with rich (logical)
languagesforreasoningaboutrelationsthatignoreuncertainty.isbookstudiestheintegration
oftheapproachestoreasoningaboutuncertaintyandreasoningaboutindividualsandrelations.
1.1 UNCERTAINTYINCOMPLEXWORLDS
Overthelast30years,ArtificialIntelligence(AI)hasevolvedfrombeingskeptical,evenhostile,
totheuseofprobabilitytoembracingprobability.Initially,manyresearcherswereskepticalabout
statisticalAIbecauseprobabilityseemedtorelyontoomanynumbersanddidnotdealwiththe
complexitiesof a world of individuals and things. But the use of probabilisticgraphical models,
exploiting probabilistic independencies, has revolutionized AI. e independencies specified in
suchmodelsarenatural,providestructurethatenablesefficientreasoningandlearning,andallow
one to model complex domains. Many AI problems arising in a wide variety of fields such as
machine learning, diagnosis, network communication, computer vision, and robotics have been
elegantlyencodedandsolvedusingprobabilisticgraphicalmodels.
Meanwhile,therehavealsobeenconsiderableadvancesinlogicalAI,whereagentsreason
aboutthe structureof complexworlds. Oneaspect of this is in the semanticweband the use of
ontologies to represent meaning in diverse fields from medicine to geology to the products in
a catalogue. Generally, there is an explosive growth in the amount of heterogeneous data that
is being collected in the business and scientific world. Example domains include biology and
chemistry,transportationsystems,communicationnetworks,socialnetworks,androbotics.Like
people,intelligentagentsshouldbeabletodealwithmanydifferenttypesofknowledge,requiring
structuredrepresentationsthatgiveamoreinformativeviewoftheAItaskathand.
Moreover,reasoningaboutindividualsandrelationsisallaboutreasoningwithregularities
and symmetries. We lump individuals into categories or classes (such as “person” or “course”)
because the individuals in a category share common properties—e.g., there are statements that
are true about all living people such as they breath, they have skin and two biological parents.
Similarly for relations, there is something in common between Sam being advised by Professor
2 1. MOTIVATION
Optimization
Cognitive
…
Science
Uncertain
Scaling
Reasoning
SAT IR
Logic Mining
Graphs and
Trees Learning
KR CV
Search DM/ML
Figure 1.1: Statistical Relational Artificial Intelligence (StarAI) combines probability, logic, and
learningandcoversmajorpartsoftheAIspectrum.
SmithandChrisbeingadvisedbyProfessorJohnson;therearestatementsaboutpublishingpa-
pers, working on a thesis and projects that are common among the “advised by” relationships.
Wewouldliketomakepredictionsabouttwopeopleaboutwhomallweknowmaybeonlytheir
advisoryrelationships.Itisthesecommonalitiesandregularitiesthatenablelanguagetodescribe
the world. Reasoning about regularities and symmetries is the foundation of logics built on the
predicatecalculus,whichallowsstatementsaboutallindividuals.
us, to deal with the real world we actually need to exploit uncertainty, independen-
cies, and symmetries and tackle a long standing goal of AI, namely unifying first-order logic—
capturing regularities and symmetries—and probability—capturing uncertainty and indepen-
dence.Predicatelogicandprobabilitytheoryarenotinconflictwitheachother,theyaresynergis-
tic.Bothextendpropositionallogic,onebyaddingrelations,individuals,andquantifiedvariables,
theotherbyallowingformeasuresoverpossibleworldsandconditionalqueries.ismayexplain
why there has been a considerable body of research in combining both of them over the last
25years,evolvingintowhathascometobecalledStatisticalRelationalArtificialIntelligence
(StarAI);seealsoFig.1.1:
thestudyanddesignofintelligentagentsthatactinworldscomposedofindividuals(objects,
things),wheretherecanbecomplexrelationsamongtheindividuals,wheretheagentscan
beuncertainaboutwhatpropertiesindividualshave,whatrelationsaretrue,whatindi-
vidualsexist,whetherdifferenttermsdenotethesameindividual,andthedynamicsofthe
world.
e basic building block of StarAI are relational probabilistic models—we use this term in the
broad sense, meaning any models that combine relations and probabilities. ey can be seen