statistical relational learning: a tutorial learning - university of

Statistical Relational Learning: A TutorialLearning: A Tutorial

Li G tLise GetoorUniversity of Maryland, College Park

AcknowledgementsThis tutorial is a synthesis of ideas of many individuals who have participated in various SRL events, workshops and classes (SRL00, SRL03, SRL04, SRL06, Dagstuhl05, Dagstuhl07, others)SRL04, SRL06, Dagstuhl05, Dagstuhl07, others)

Dagstuhl April 2007

Hendrik Blockeel, Mark Craven, James Cussens, Bruce D’Ambrosio, Luc De Raedt, Tom Dietterich, Pedro Domingos, Saso Dzeroski, Peter Flach, Rob Holte, Manfred Jaeger, David Jensen, Kristian Kersting, Heikki Mannila, Andrew McCallum, Tom

Dagstuhl April 2007

Mitchell, Ray Mooney, Stephen Muggleton, Kevin Murphy, Jen Neville, David Page, Avi Pfeffer, Claudia Perlich, David Poole, Foster Provost, Dan Roth, Stuart Russell, Taisuke Sato, Jude Shavlik, Ben Taskar, Lyle Ungar and many others

RoadmapSRL: What is it?SRL Tasks & ChallengesSRL Tasks & Challenges4 SRL ApproachesApplications and Future directionspp cat o s a d utu e d ect o s

Why SRL?Traditional statistical machine learning approaches assume:

A random sample of homogeneous objects from single relation

Traditional ILP/relational learning approaches assume:No noise or uncertainty in data

Real world data sets:Real world data sets:Multi-relational, heterogeneous and semi-structuredNoisy and uncertain

Statistical Relational Learning:newly emerging research area at the intersection of research in social network and link analysis, hypertext and web mining, graph mining, relational learning and inductive logic programmingg g g p g g

Sample Domains:web data, bibliographic data, epidemiological data, communication data customer networks collaborative filtering trust networksdata, customer networks, collaborative filtering, trust networks, biological data, natural language, vision

What is SRL?Three views…

View 1: Alphabet Soup

View 2: Representation Soup

Hierarchical Bayesian Model + Relational R t tiRepresentation

Logic Add probabilities StatisticalLogic Add probabilities

Add relationsProbabilities

Relational

Learning

View 3: Data Soup

Training Data Test Data

GoalsBy the end of this tutorial, hopefully, you will be:

1. able to distinguish among different SRL tasksg g2. able to represent a problem in one of several SRL

representations3. excited about SRL research problems and practical

applications

SRL TasksTasks

Object ClassificationjObject Type PredictionLink Type PredictionPredicting Link ExistencePredicting Link ExistenceLink Cardinality EstimationEntity ResolutionGroup Detection Subgraph DiscoveryMetadata Miningg

Object Prediction

Object ClassificationPredicting the category of an object based on itsPredicting the category of an object based on its attributes and its links and attributes of linked objectse.g., predicting the topic of a paper based on the words used in the paper the topics of papers it cites theused in the paper, the topics of papers it cites, the research interests of the author

Object Type PredictionObject Type PredictionPredicting the type of an object based on its attributes andits links and attributes of linked objectse g predict the venue type of a publication (conferencee.g., predict the venue type of a publication (conference, journal, workshop) based on properties of the paper

Link PredictionLink Classification

Predicting type or purpose of link based on properties of the participating objectsparticipating objects e.g., predict whether a citation is to foundational work, background material, gratuitous PC reference

Predicting Link ExistencePredicting whether a link exists between two objectse.g. predicting whether a paper will cite another paper

Link Cardinality EstimationyPredicting the number of links to an object or predicting the number of objects reached along a path from an objecte.g., predict the number of citations of a paperg , p p p

More complex prediction tasksGroup Detection

Predicting when a set of entities belong to the same group based on clustering both object attribute values and link structureon clustering both object attribute values and link structuree.g., identifying research communities

Entity ResolutionEntity ResolutionPredicting when a collection of objects are the same, based on their attributes and their links (aka: record linkage, identity uncertainty)

di ti h t it ti f i t the.g., predicting when two citations are referring to the same paper.

Predicate InventionPredicate InventionInduce a new general relation/link from existing links and pathse.g., propose concept of advisor from co-author and financial supportpp

Subgraph Identification, Metadata Mapping

SRL ChallengesCollective ClassificationCollective Consolidation Logical vs. Statistical dependenciesFeature Construction – aggregation, selectionFlexible and Decomposable Combining RulesFlexible and Decomposable Combining RulesInstances vs. ClassesEffective Use of Labeled & Unlabeled DataEffective Use of Labeled & Unlabeled DataLink PredictionClosed vs. Open Worldp

Challenges common to any SRL approachl!Bayesian Logic Programs Markov Logic Networks Probabilistic Relational Models Bayesian Logic Programs, Markov Logic Networks, Probabilistic Relational Models,

Relational Markov Networks, Relational Probability Trees, Stochastic Logic Programming to name a few

Four SRL ApproachesDirected Approaches

Rule-based Directed ModelsFrame-based Directed Models

Undirected ApproachesFrame-based Undirected ModelsRule-based Undirected Models

P i L A h ( fi !)Programming Language Approaches (oops, five!)

Emphasis in Different ApproachesRule-based approaches focus on facts

what is true in the world?h t f t d th f t d d ?what facts do other facts depend on?

Frame-based approaches focus on objects and relationshipswhat types of objects are there, and how are they related to each th ?other?

how does a property of an object depend on other properties (of the same or other objects)?

Directed approaches focus on causal interactionsDirected approaches focus on causal interactionsUndirected approaches focus on symmetric, non-causal interactionsP i l h fProgramming language approaches focus on processes

how is the world generated?how does one event influence another event?


BN TutorialRule-based Directed ModelsFrame-based Directed Models

Undirected ApproachesMarkov Network TutorialRule-based Undirected ModelsF b d U di t d M d lFrame-based Undirected Models

Bayesian Networks

Good Writer Smart SW P(Q| W S)

conditional probability table (CPT)

Good Writer0.6 0.4

w 0 3 0 7

sw

s

SW P(Q| W, S)

QualityReviewer

Mood

w

s

w

0.3 0.7

0 1 0 90.4 0.6

s

s

w

nodes = domain variables

s 0.1 0.9w

AcceptedReview Length

edges = direct causal influence

Network structure encodes conditional independencies:I(Review-Length , Good-Writer | Reviewer-Mood)

BN Semantics

conditional f ll j i tW S

conditionalindependenciesin BN structure

+ localCPTs

full jointdistribution

over domain=M Q

AL AL

),,,,,( alqmswP =

),()|(),|()|()()(),,,,,(

qmaPmlPswqPwmPsPwPalqmswP

|Compact & natural representation:

nodes ≤ k parents ⇒ O(2k n) vs. O(2n) paramsnatural parameters

Reasoning in BNsFull joint distribution answers any query

P(event | evidence)(e e t | e de ce)

Allows combination of different types of reasoning:yp gCausal: P(Reviewer-Mood | Good-Writer)Evidential: P(Reviewer-Mood | not Accepted)Intercausal: P(Reviewer-Mood | not Accepted, Quality)

W SW

M

S

Q

AL

Variable Elimination

To compute ∑= alqmswPaP )()(o co pute ∑lqmsw

alqmswPaP,,,,

),,,,,()(

factors

∑ qmaPmlPswqPwmPsPwP )|()|()|()|()()(

factors

∑lqmsw

qmaPmlPswqPwmPsPwP,,,,

),|()|(),|()|()()(

mood good writerpissypissy true

false01 A factor is a function from

values of variables togoodgood

falsetrue

0.70.3

values of variables to positive real numbers


To compute ∑= alqmswPaP )()(To compute ∑lqmsw

alqmswPaP,,,,

),,,,,()(

∑ ∑ qmaPmlPswqPwmPsPwP )|()|()|()|()()(∑ ∑qmsw l

qmaPmlPswqPwmPsPwP,,,

),|()|(),|()|()()(


To compute ∑= alqmswPaP )()(To compute ∑=lqmsw

alqmswPaP,,,,

),,,,,()(

∑∑ lPPPPPP )|()|()|()|()()( ∑∑lqmsw

mlPqmaPswqPwmPsPwP )|(),|(),|()|()()(,,,

sum out l



alqmswPaP,,,,

),,,,,()(

)()|()|()|()()( mfqmaPswqPwmPsPwP∑ )(),|(),|()|()()(,,,

mfqmaPswqPwmPsPwPqmsw

∑ 1

new factornew factor


To compute ∑= alqmswPaP )()(To compute ∑=lqmsw

alqmswPaP,,,,

),,,,,()(

)|()|()()()|()( sqPmPPmfqmaPsP ∑∑ ),|()|()()(),|()(,,

swqPwmPwPmfqmaPsPwqms∑∑ 1

multiply factors togethery gthen sum out w



alqmswPaP,,,,

),,,,,()(

)()()|()( sqmfmfqmaPsP∑ ),,()(),|()(,,

sqmfmfqmaPsPqms

21∑new factornew factor



alqmswPaP,,,,

),,,,,()(

)(aP )(aP

Other Inference AlgorithmsExact

Junction Tree [Lauritzen & Spiegelhalter 88][ & p g ]Cutset Conditioning [Pearl 87]

ApproximateLoopy Belief Propagation [McEliece et al 98]Likelihood Weighting [Shwe & Cooper 91]Markov Chain Monte Carlo [eg MacKay 98]

• Gibbs Sampling [Geman & Geman 84]• Metropolis-Hastings [Metropolis et al 53 Hastings 70]Metropolis Hastings [Metropolis et al 53, Hastings 70]

Variational Methods [Jordan et al 98]

Learning BNs

Parameters only Structure andParameters onlyParameters

Complete Data Easy: counting Structure search

EM [Dempster et al 77]Incomplete Data

EM [Dempster et al 77]or gradient descent

[Russell et al 95]

Structural EM [Friedman 97]

[ ]

See [Heckerman 98] for a general introduction

BN Parameter EstimationAssume known dependency structure GGoal: estimate BN parameters θp

entries in local probability models,

)][Pa|( uXxXP ===θ

θ is good if it’s likely to generate observed data.

)][Pa|(, uXxXPuxθ

),|(log),:( GDPGDl θθ =

MLE Principle: Choose θ∗ so as to maximize lAlt ti i t iAlternative: incorporate a prior

Learning With Complete DataFully observed data: data consists of set of instances, each with a value for all BN variables

With fully observed data, we can compute swqN ,,= number of instances with , and

and similarly for other counts

q ,,q w s

We then estimateN

sw

swqswq N

NswqP

,

,,,, ),|( ==θ

Dealing w/ missing valuesCan’t computeBut can use Expectation Maximization (EM)

swqN ,,

But can use Expectation Maximization (EM) 1. Given parameter values, can compute expected

counts: ∑= iiiiswq swqPNE ,, )evidence|,,(][

2. Given expected counts, estimate parameters:

∑i

q instances

,,

][NE

this requires BN inference

Begin with arbitrary parameter values][][

),|(,

,,,,

sw

swqswq NE

NEswqP ==θ

Begin with arbitrary parameter valuesIterate these two stepsConverges to local maximum of likelihoodConverges to local maximum of likelihood

Structure searchBegin with an empty networkConsider all neighbors reached by a search operator g y pthat are acyclic

add an edgeremove an edgeremove an edgereverse an edge

For each neighborcompute ML parameter values

*

*sθ

compute score(s) =

Choose the neighbor with the highest score

)(log),|(log * sPsDP s +θ

Choose the neighbor with the highest scoreContinue until reach a local maximum

Mini-BN Tutorial SummaryRepresentation – probability distribution factored according to the BN DAGgInference – exact + approximateLearning – parameters + structure



Undirected ApproachesMarkov Network TutorialFrame-based Undirected ModelsR l b d U di t d M d lRule-based Undirected Models

Directed Rule-based FlavorsGoldman & Charniak [93]Breese [92][ ]Probabilistic Horn Abduction [Poole 93]Probabilistic Logic Programming [Ngo & Haddawy 96]Relational Bayesian Networks [Jaeger 97]Bayesian Logic Programs [Kersting & de Raedt 00]Stochastic Logic Programs [Muggleton 96]Stochastic Logic Programs [Muggleton 96]PRISM [Sato & Kameya 97]CLP(BN) [Costa et al. 03]Logical Bayesian Networks [Fierens et al 04, 05]etc.

Intuitive ApproachIn logic programming,

accepted(P) :- author(P,A), famous(A).

meansFor all P,A if A is the author of P and A is famous, then P

is acceptedThis is a categorical inferenceB t thi t b t i llBut this may not be true in all cases

Fudge FactorsUse

accepted(P) :- author(P,A), famous(A). (0.6)This means

For all P,A if A is the author of P and A is famous, then Pis accepted with probability 0 6is accepted with probability 0.6

But what does this mean when there are otherBut what does this mean when there are other possible causes of a paper being accepted?e.g. accepted(P) :- high_quality(P). (0.8)

Intuitive Meaningaccepted(P) :- author(P,A), famous(A). (0.6)

meansFor all P,A if A is the author of P and A is famous, then P

is accepted with probability 0.6, provided no other possible cause of the paper being accepted holdspossible cause of the paper being accepted holds

If more than one possible cause holds a combiningIf more than one possible cause holds, a combining rule is needed to combine the probabilities

Meaning of DisjunctionIn logic programming

accepted(P) :- author(P,A), famous(A). accepted(P) :- high_quality(P).

means For all P,A if A is the author of P and A is famous, or if P

is high quality, then P is accepted

Probabilistic DisjunctionNow

accepted(P) :- author(P,A), famous(A). (0.6)accepted(P) : high quality(P) (0 8)accepted(P) :- high_quality(P). (0.8)

meansFor all P,A, if (A is the author of P and A is famous , , (

successfully cause P to be accepted) or (P is high quality successfully causes P to be accepted), then Pis accepted.p

If A is the author of P and A is famous, they successfully cause P to be accepted with probability 0.6.

If P is high quality it successfully causes P to beIf P is high quality, it successfully causes P to be accepted with probability 0.8.

All causes act independently to produce effect (causal independence)p y p ( p )Leak probability: effect may happen with no cause

e.g. accepted(P). (0.1)

Computing ProbabilitiesWhat is P(accepted(p1)) given that Alice is an author and Alice is famous, and that the paper is p phigh quality, but no other possible cause is true?

1 fail) causes possible true all(succeeds) causetrueoneleast P(at

−==

PP

92801018016011 .).)(.)(.(

))(-(1-1 successcauses possible true

=−−−−=

= ∏ ip i

))()((

leak

Combination Rules

Other combination rules are possiblee.g., max

)(maxeffect)( successibltipP

i=

In our case,P(accepted(p1)) = max {0 6 0 8 0 1} = 0 8

causespossibletrue i

P(accepted(p1)) = max {0.6,0.8,0.1} = 0.8Harder to interpret in terms of logic program

KBMCKnowledge-Based Model Construction (KBMC) [Wellman et al. 92, Ngo & Haddawy 95]M th d f ti l b bilitiMethod for computing more complex probabilitiesConstruct a Bayesian network, given a query Qand evidence Eand evidence E

query and evidence are sets of ground atoms, i.e., predicates with no variable symbols

th ( 1 li )• e.g. author(p1,alice)

Construct network by searching for possible proofs of the query and the variablesp q yUse standard BN inference techniques on constructed network

KBMC Example

smart(alice). (0.8)smart(bob). (0.9)author(p1,alice). (0.7)author(p1,bob). (0.3)(p , ) ( )high_quality(P) :- author(P,A), smart(A). (0.5)high_quality(P). (0.1)accepted(P) : high quality(P) (0 9)accepted(P) :- high_quality(P). (0.9)

Query is accepted(p1)Query is accepted(p1).Evidence is smart(bob).

Backward ChainingStart with evidence variable smart(bob)

smart(bob)

Backward ChainingRule for smart(bob) has no antecedents – stop

backward chainingg

smart(bob)

Backward ChainingBegin with query variable accepted(p1)

smart(bob)

accepted(p1)

Backward Chaining

Rule for accepted(p1) has antecedent high_quality(p1)dd hi h lit ( 1) t t k d k t fadd high_quality(p1) to network, and make parent of accepted(p1)

smart(bob)

high_quality(p1)

accepted(p1)

Backward ChainingAll of accepted(p1)’s parents have been found –

create its conditional probability table (CPT)p y ( )

smart(bob)

high_quality(p1)high_quality(p1) accepted(p1)

accepted(p1) hq

hq 0.9 0.1

0 1

Backward Chaininghigh_quality(p1) :- author(p1,A), smart(A) has two

groundings: A=alice and A=bobg g

smart(bob)

high_quality(p1)

accepted(p1)

Backward ChainingFor grounding A=alice, add author(p1,alice) and

smart(alice) to network, and make parents ofsmart(alice) to network, and make parents of high_quality(p1)

smart(bob)smart(alice)

author(p1,alice)

high_quality(p1)

accepted(p1)

Backward ChainingFor grounding A=bob, add author(p1,bob) to network.

smart(bob) is already in network. Make both ( ) yparents of high_quality(p1)


author(p1,alice) author(p1,bob)

high_quality(p1)

accepted(p1)

Backward ChainingCreate CPT for high_quality(p1) – make noisy-or



high_quality(p1)

accepted(p1)

Backward Chainingauthor(p1,alice), smart(alice) and author(p1,bob)

have no antecedents – stop backward chainingp g



high_quality(p1)

accepted(p1)

Backward Chainingassert evidence smart(bob) = true, and compute P(accepted(p1) | smart(bob) = true)(accepted(p ) | s a t(bob) t ue)

tsmart(bob)smart(alice)

true


high_quality(p1)

accepted(p1)

Backward Chaining on Both Query and EvidenceQuery and Evidence

Necessary, if query and evidence have common ancestor

Ancestor

S ffi i t P(Q | E id ) b t d

Query Evidence

Sufficient. P(Query | Evidence) can be computed using only ancestors of query and evidence nodesnodes

unobserved descendants are irrelevant

The Role of ContextContext is deterministic knowledge known prior to the network being constructedg

May be defined by its own logic programIs not a random variable in the BN

Used to determine structure of the constructed BNIf a context predicate P appears in the body of a rule R only backward chain on R if P is trueR, only backward chain on R if P is true

Context exampleSuppose author(P,A) is a context predicate,

author(p1,bob) is true, and author(p1,alice) cannot (p , ) (p , )be proven from deterministic KB (and is therefore false by assumption)

Network is

N th ( 1 b b) dsmart(bob) No author(p1,bob) nodebecause it is a context predicate

high_quality(p1) No smart(alice) nodebecause author(p1,alice) is false

accepted(p1)

SemanticsAssumption: no cycles in resulting BN

If there are cycles, cannot interpret BN as definition y , pof joint probability distribution

Assuming BN construction process terminates, conditional probability of any query given any evidence is defined by the BN.Somewhat unsatisfying becauseSomewhat unsatisfying because

1. meaning of program is query dependent (depends on constructed BN))

2. meaning is not stated declaratively in terms of program but in terms of constructed network i t dinstead

Disadvantages of Approach

Up until now, ground logical atoms have been d i bl irandom variables ranging over T,Fcumbersome to have a different random variable for lead author(p1,alice), lead author(p1,bob)for lead_author(p1,alice), lead_author(p1,bob)and all possible values of lead_author(p1,A)

worse, since lead_author(p1,alice) and lead_author(p1,bob) are different random variables, it is possible for both to be true at thevariables, it is possible for both to be true at the same time

Bayesian Logic Programs [Kersting and de Raedt][Kersting and de Raedt]

Now, ground atoms are random variables with gany range (not necessarily Boolean)

now quality is a random variable, with values high medi m lohigh, medium, low

Any probabilistic relationship is allowedAny probabilistic relationship is allowedexpressed in CPT

Semantics of program given once and for allnot query dependentnot query dependent

Meaning of Rules in BLPsaccepted(P) :- quality(P).

meansmeans“For all P, if quality(P) is a random variable, then accepted(P) is a random variable”

Associated with this rule is a conditional probability table (CPT) that specifies the probability distribution over accepted(P) for any possible value ofquality(P)quality(P)

Combining Rules for BLPs

accepted(P) :- quality(P).accepted(P) : quality(P).accepted(P) :- author(P,A), fame(A).

Before, combining rules combined individual probabilities with each otherp

noisy-or and max rules easy to interpretNow, combining rules combine entire CPTs

Semantics of BLPsRandom variables are all ground atoms that have finite proofs in logic programsp g p g

assumes acyclicityassumes no function symbols

Can construct BN over all random variablesparents derived from rulesCPTs derived using combining rules

Semantics of BLP: joint probability distribution over all random variablesall random variables

does not depend on queryInference in BLP by KBMCInference in BLP by KBMC

An IssueHow to specify uncertainty over single-valued relations?Approach 1: make lead_author(P) a random variable taking values bob alice etcvalues bob, alice etc.

we can’t say accepted(P) :- lead_author(P), famous(A)because A does not appear in the rule head or in a previous term in the bodyprevious term in the body

Approach 2: make lead_author(P,A) a random variable with values true, false

i t th bl ith th i t itiwe run into the same problems as with the intuitive approach (may have zero or many lead authors)

Approach 3: make lead_author a functionsay accepted(P) :- famous(lead_author(P))need to specify how to deal with function symbols and uncertainty over themy

First-Order Variable Elimination[Poole 03, Braz et al 05] Generalization of variable elimination to firstGeneralization of variable elimination to first order domainsReasons directly about first-order variables, y ,instead of at the ground levelAssumes that the size of the population for each type of entity is known

Learning Rule Parameters

[Koller & Pfeffer 97, Sato & Kameya 01] P bl d fi itiProblem definition:

Given a skeleton rule base consisting of rules without uncertainty parameterswithout uncertainty parametersand a set of instances, each with

• a set of context predicates• observations about some random variables

Goal: learn parameter values for the rules that maximize the likelihood of the datamaximize the likelihood of the data

Basic Approach1. Construct a network BNi for each instance i using

KBMC, backward chaining on all the observed gvariables

2. Expectation Maximization (EM)• exploit parameter sharing

Parameter SharingIn BNs, all random variables have distinct CPTs

only share parameters between different instances, not different random variables

In logical approaches, an instance may contain many objects of the same kindmany objects of the same kind

multiple papers, multiple authors, multiple citationsParameters are shared within instances

same parameters used across different papers, authors, citations

Parameter sharing allows faster learning andParameter sharing allows faster learning, and learning from a single instance

Rule Parameters & CPT EntriesIn principle, combining rules produce complicated relationship between model parameters and CPT p pentriesWith a decomposable combining rule, each node

fis derived from a single ruleMost natural combining rules are decomposable

• e g noisy or decomposes into set of ands followed by or• e.g. noisy-or decomposes into set of ands followed by or

Parameters and CountsEach time a node is derived from a rule r, it provides one experiment to learn about the parameters

i t d ithassociated with r

Each such node should therefore make a separateEach such node should therefore make a separate contribution to the count for those parameters

: the parameter associated with P(X=x|Parents[X]=u) when rule r applies

rux ,θ

: the number of times a node has value x and its parents have value u when rule r applies

ruxN ,p pp

EM With Parameter SharingGiven parameter values, compute expected counts:

)|],(][ ,i

i X

rux uXxXPNE evidenceParents[

instances === ∑ ∑

where the inner sum is over all nodes derived from rule r in BNi

Given expected counts, estimate:

][][ ,

, ru

ruxr

ux NENE

=θ

Iterate these two steps

][ u

Learning Rule Structure

[Kersting and De Raedt 02] P bl d fi itiProblem definition:

Given a set of instances, each with • context predicatescontext predicates• observations about some random variables

Goal: learn • a skeleton rule base consisting of rules and parameter

values for the rules

Generalizes BN structure learningGe e a es s uc u e ea gdefine legal modelsscoring function same as for BNdefine search operators

Legal ModelsHypothesis space consists of all rule sets using given predicates, together with parameter valuesg p g pA legal hypothesis:

is logically valid: rule set does not draw false conclusions for any data casesthe constructed BN is acyclic for every instance

Search operatorsAdd a constant-free atom to the body of a single clauseRemove a constant-free atom from the body of a single clause

accepted(P) :- author(P,A).accepted(P) :- quality(P).

accepted(P).t d(P) lit (P)

delete add

accepted(P) :- quality(P).

accepted(P) :- author(P,A), famous(A).accepted(P) :- quality(P)accepted(P) :- quality(P).

Summary: Directed Rule-based ApproachesApproaches

Provide an intuitive way to describe how one fact d d th f tdepends on other factsIncorporate relationships between entitiesGeneralizes to many different situationsGeneralizes to many different situations

Constructed BN for a domain depends on which objects exist and what the known relationships are j pbetween them (context)

Inference at the ground level via KBMCor lifted inference via FOVE

Both parameters and structure are learnable



Undirected ApproachesMarkov Network TutorialFrame-based Undirected ModelsR l b d U di t d M d lRule-based Undirected Models

Frame-based ApproachesProbabilistic Relational Models (PRMs)

Representation & Inference [Koller & Pfeffer 98, p & [ & ,Pfeffer, Koller, Milch &Takusagawa 99, Pfeffer 00] Learning [Friedman et al. 99, Getoor, Friedman, K ll & T k 01 & 02 G t 01]Koller & Taskar 01 & 02, Getoor 01]

Probabilistic Entity Relation Models (PERs)Probabilistic Entity Relation Models (PERs)Representation [Heckerman, Meek & Koller 04]


BN TutorialR l b d Di t d M d lRule-based Directed ModelsFrame-based Directed Models

• PRMs w/ Attribute UncertaintyInference in PRMs• Inference in PRMs

• Learning in PRMs• PRMs w/ Structural Uncertainty

• PRMs w/ Class Hierarchies

Undirected ApproachesMarkov Network TutorialFrame-based Undirected ModelsRule-based Undirected Models

Relational Schema

AuthorG d W it

ReviewGood Writer Mood

LengthSmart

PaperQuality

Author ofHas ReviewAccepted

Describes the types of objects and relations in the database

Probabilistic Relational Model

Author ReviewSmart

Length

Mood

Good Writer

Smart

g

PaperPaper

Quality

Accepted


Author ReviewSmart

Length

Mood

Good Writer

Smart

g

PaperPaper

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⎠⎜⎜⎜

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Author ReviewSmart

Length

Mood

Good Writer

Smart

g

PaperPaper

Quality

80209.01.0,

,

tfff

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30704.06.08.02.0

,,

ttfttf

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Relational Skeleton

P i KAuthor A1

Paper P1Author: A1Review: R1

Review R1

Paper P2

Primary Keys

Review R2Author A2


P P3 Review R2Paper P3Author: A2Review: R2 Foreign Keys

Review R2

Fixed relational skeleton σ:set of objects in each classrelations between them

PRM w/ Attribute UncertaintyAuthor A1

Paper P1Author: A1Review: R1 Review R1

SmartMoodQuality

Paper P2

Good Writer Length

Mood

Accepted

Review R2Author A2 Author: A1

Review: R2

Length

MoodQuality

AcceptedGood Writer

Smart


Review R3

Accepted

Review: R2Quality

AcceptedLength

Mood

PRM defines distribution over instantiations of attributes

A Portion of the BN

P2.Quality r2.Mood

9010,

ffP(A | Q, M)MQPissyLow

9010,

ffP(A | Q, M)MQ

P2.Accepted

4.06.08.02.09.01.0

,,,

fttfff

4.06.08.02.09.01.0

,,,

fttfff

P3 Accepted

P3.Quality3.07.04.06.0

,,ttft

3.07.04.06.0

,,ttft

r3.Mood

P3.Accepted

A Portion of the BN

P2.Quality r2.MoodPissyLow

9.01.0,,

ffP(A | Q, M)MQ

P2.Accepted

4.06.08.02.09.01.0

,,,

fttfff

P3 Accepted

P3.Quality r3.MoodHigh 3.07.0, ttPissy

P3.Accepted

PRM: Aggregate Dependencies

Mood

Paper

Quality

Review

Length

Quality

Accepted

Review R1

Length

Mood

Review R2

MoodPaper P1

LengthReview R3

MoodAccepted

Quality

Length

PRM: Aggregate Dependencies

Mood

Paper

Quality

Review

Length

Quality

Accepted

Review R19.01.0,

,ff

P(A | Q, M)MQ

Length

Mood

Review R2

MoodPaper P1 30704.06.08.02.0

,,,

ttfttf

LengthReview R3

MoodAccepted

Quality

mode

3.07.0, tt

sum, min, max, avg, mode, count

Length

PRM with AU SemanticsAuthor

Paper AuthorA1 Paper

R i

ReviewR1

Review PaperP2

P1

ReviewR3

ReviewR2Author

A2

Paper

PRM relational skeleton σ+ =

PaperP3

probability distribution over completions I:

)).(|.(),,|( ,.

AxparentsAxPP Sx Ax

σσ

σ ∏ ∏∈

=ΘSI

AttributesObjects

PRM InferenceSimple idea: enumerate all attributes of all objectsConstruct a Bayesian network over all the attributesConstruct a Bayesian network over all the attributes

Inference Example

Review

Review

R1Paper

P1

Skeleton

R2Author

A1Review

R3

ReviewR4

R3Paper

P2

R4

Query is P(A1 good writer)Query is P(A1.good-writer)Evidence is P1.accepted = T, P2.accepted = T

PRM Inference: Constructed BN

A1.Smart

P1.Quality

A1.Good WriterQ y

R1.Mood R2.Mood

P1 AcceptedR1.Length R2.Length P2.Quality

P1.AcceptedR3.Mood R4.Mood

P2.AcceptedR3.Length R4.Length

PRM InferenceProblems with this approach:

constructed BN may be very largey y gdoesn’t exploit object structure

Better approach:reason about objects themselvesreason about whole classes of objects

In particular, exploit:reuse of inference

l ti f bj tencapsulation of objects

PRM Inference: Interfaces

A1.SmartVariables pertainingto R2: inputs and i t l tt ib t

A1.Good Writer

internal attributes

R2.MoodR1.Mood

P1.QualityQ yR2.LengthR1.Length

P1.Accepted

PRM Inference: Interfaces

A1.SmartInterface: imported and exported

A1.Good Writer

exportedattributes

R2.MoodR1.Mood

P1.QualityQ yR2.LengthR1.Length

P1.Accepted

PRM Inference: Encapsulation

A1.Smart R1 and R2 areencapsulated

P1.QualityA1.Good Writer inside P1

Q y

R1.Mood R2.Mood




PRM Inference: Reuse

A1.Smart

P1.QualityA1.Good Writer

Q y

R1.Mood R2.Mood




Structured Variable Elimination

Author 1

A1.Smart

A1 Good WriterA1.Good Writer

Paper-1

Paper-2

P 1Structured Variable Elimination

A1.Smart

Paper 1

A1.Good Writer

P1 Quality

Review-2Review-1

P1.Quality

P1.Accepted


A1.Smart

Paper 1

A1.Good Writer

Review-2Review-1

P1 QualityP1.Quality

P1.Accepted


Review 2

A1.Good Writer

R2 MoodR2.Mood

R2 L hR2.Length


Review 2

A1.Good Writer

R2 MoodR2.Mood


A1.Smart

Paper 1

A1.Good Writer

Review-2Review-1


P1.Accepted


A1.Smart

Paper 1

A1.Good Writer

Review-1 R2.Mood


P1.Accepted

Structured Variable EliminationP 1

A1.Smart

Paper 1

A1.Good Writer

Review-1 R2.Mood


P1.Accepted


A1.Smart

Paper 1

A1.Good Writer

R2.MoodR1.Mood


P1.Accepted


A1.Smart

Paper 1

A1.Good Writer

R2.MoodR1.Mood


P1.Accepted True


A1.Smart

Paper 1

A1.Good Writer


Author 1

A1.Smart


Paper-1

Paper-2


Author 1

A1.Smart


Paper-2


Author 1

A1.Smart



Author 1


Benefits of SVEStructured inference leads to good elimination orderings for VEg

interfaces are separators• finding good separators for large BNs is very hard

therefore cheaper BN inferenceReuses computation wherever possible

Limitations of SVEDoes not work when encapsulation breaks down

ReviewerP R2

AuthorA1

PaperP1

ReviewerR3Paper

P2

R3 is notencapsulatedinside P2

But when we don’t have specific information about the

ReviewerR4

P2

connections between objects, we can assume that encapsulation holds

i.e., if we know P1 has two reviewers R1 and R2 but they are not ynamed instances, we assume R1 and R2 are encapsulated

Cannot reuse computation when different objects have different evidenceevidence

Learning PRMs w/ AUDatabase

Paper

Author

p

Review

A th

Paper

Author

Review

• Parameter estimation• Structure selection

RelationalSchema

ML Parameter Estimation

PaperQ lit

ReviewMoodLength

, P(A | Q, M)MQQuality

Accepted

Length

,,tfff ?

???

APMRQPN .,.,.θ∗

,,ttft ?

???

MRQP

Q

N .,.

,,θ∗ =

APMRQPN .,.,.where is the number of accepted, l litlow quality papers whose reviewer was in a poor mood

ML Parameter Estimation

PaperQ lit

ReviewMoodLength

, P(A | Q, M)MQQuality

Accepted

Length

,,tfff ?

???

APMRQPN .,.,.θ∗

,,ttft ?

???

MRQP

Q

N .,.

,,θ∗ =

Q f

Count

Query for counts:

Review PaperπCount table table

AcceptedPMoodR

QualityP

..

.π

Structure Selection

Idea:d fi i f tidefine scoring function do local search over legal structures

Key Components:legal models scoring modelssearching model space

Structure Selection

Idea:d fi i f tidefine scoring function do local search over legal structures

Key Components:» legal models

scoring modelssearching model space

Legal Models

PRM defines a coherent probability model over a skeleton σ if the dependencies between object

th f

skeleton σ if the dependencies between object attributes is acyclic

Paperauthor-of

ResearcherProf. Gump

Reputation

P1Accepted

yes PaperP2p

highP2

Accepted yes

sum

How do we guarantee that a PRM is acyclic for every skeleton?for every skeleton?

Attribute StratificationPRM

dependency dependencygraphstructure S graph

Paper AcceptedPaper.Acceptedif Researcher.Reputationdepends directly on Paper.Accepted

Researcher.Reputation

Attribute stratification:dependency graph acyclic ⇒ acyclic for any σ

Attribute stratification:

Algorithm more flexible; allows certainAlgorithm more flexible; allows certain cycles along guaranteed acyclic relations

Structure Selection

Idea: d fi i f tidefine scoring function do local search over legal structures

Key Components:legal models

» scoring models – same as BNsearching model space

Structure Selection

Idea: d fi i f tidefine scoring function do local search over legal structures

Key Components:legal models scoring models

» searching model space

Searching Model SpacePhase 0: consider only dependencies within a class

Author ReviewPaper

ReviewAuthor Paper

U)( BRARParentsPotential U)(.

.).(RattributesedescriptivBR

BRARParentsPotential−∈

=−

ReviewAuthor Paper

Phased Structure SearchPhase 1: consider dependencies from “neighboring”

classes, via schema relations

ReviewPaperAuthor

ReviewAuthor Paper

U)( CSARParentsPotential U<> )(.

.).(SRattributesedescriptivCS

CSARParentsPotential−∈

=−

ReviewAuthor Paper

Phased Structure SearchPhase 2: consider dependencies from “further”

classes, via relation chains

ReviewAuthor Paper

ReviewAuthor Paper

U)( DTARParentsPotential U<><> )(.

.).(TSRattributesedescriptivDT

DTARParentsPotential−∈

=−

ReviewAuthor Paper

Reminder: PRM w/ AU SemanticsAuthor

Paper AuthorA1 Paper

R i

ReviewR1

Review PaperP2

P1

ReviewR3

ReviewR2Author

A2

Paper

PRM relational skeleton σ+ =

PaperP3

probability distribution over completions I:

)).(|.(),,|( ,.

AxparentsAxPP Sx Ax

σσ

σ ∏ ∏∈

=ΘSI

AttributesObjects

Kinds of structural uncertaintyHow many objects does an object relate to?

how many Authors does Paper1 have?Which object is an object related to?

does Paper1 cite Paper2 or Paper3?Which class does an object belong to?

is Paper1 a JournalArticle or a ConferencePaper?D bj t t ll i t?Does an object actually exist?Are two objects identical?

Structural UncertaintyMotivation: PRM with AU only well-defined when the skeleton structure is knownMay be uncertain about relational structure itselfConstruct probabilistic models of relational structure that capture structural uncertaintythat capture structural uncertaintyMechanisms:

Reference uncertaintyReference uncertaintyExistence uncertainty Number uncertaintyNumber uncertaintyType uncertaintyIdentity uncertaintyy y

Citation Relational Schema

AuthorInstitutionInstitutionResearch Area

Wrote

PaperPaperTopicp

TopicWord1Word2

pWord1

…Word2Cites

Citi

WordN… WordNCiting

PaperCited Paper

Attribute Uncertainty

AuthorInstitution P( Institution |

Research Area

P( Institution | Research Area)

Wrote P( Topic | Paper Author Research Area

PaperTopic

Paper.Author.Research Area

Word1 WordN...

P( WordN | Topic)

Reference Uncertainty

Bibliography

`1. -----2. -----

??

Scientific Paper

2. 3. -----?

Scientific Paper

Document CollectionDocument Collection

PRM w/ Reference Uncertainty

Paper Paper

CitesCiting

pTopicWords

PaperTopicWordsCiting

Cited

Dependency model for foreign keys

N ï A h lti i l i kNaïve Approach: multinomial over primary key• noncompact• limits ability to generalize• limits ability to generalize

Reference Uncertainty ExamplePaper

P5Topic

PaperP4

T i

PaperP3

PaperM2Paper

PaperP5

PaperC1Topic AI

Topic AI

Topic AI

M2Topic

AI

PaperP1

Topic Theory

P5Topic

AIPaper

P3

P4Topic Theory

PaperP2

Topic Theory

PaperP1

C1

P3Topic

AITopic Theory

Paper.Topic = AI Paper.Topic = TheoryC2

p p y

CitesCitingCited

C1 C23.0 7.0

Cited

Reference Uncertainty ExamplePaper

P5Topic

PaperP4

T i

PaperP3

PaperM2Paper

PaperP5

PaperC1Topic AI

Topic AI

Topic AI

M2Topic

AI

PaperP1

Topic Theory

P5Topic

AIPaper

P3

P4Topic Theory

PaperP2

Topic Theory

PaperP6

C1

P3Topic

AITopic Theory

Paper.Topic = AI Paper.Topic = TheoryC2

p p y

PaperTopic C1 C2TopicCites

CitingCited

TopicWords

C1 C21.0 9.0

Topic

99.0 01.0Theory

AICited 99.0 01.0

Introduce Selector RVs

Cites1.Selector P2.Topic

Cites1.Cited P3.Topic

P1.Topic P4.Topic

Cites2 SelectorP5.Topic

Cites2 Cited

Cites2.Selector

P6.TopicCites2.Cited

Introduce Selector RV whose domain is {C1 C2}Introduce Selector RV, whose domain is {C1,C2}The distribution over Cited depends on all of the topics, and the selector

PRMs w/ RU Semantics

Paper Paper PaperP5

PaperP2 PaperPaper

P2

CitesCitedCiti

TopicWords

TopicWords

P5Topic

AI

PaperP4Topic Theory

P2Topic TheoryPaperP3Topic AI

PaperP1

Topic ???

P5Topic

AI

PaperP4Topic Theory


PaperP1

Topic

RegReg

RegRegCitesCiting

PRM RU

??? ???gRegCites

entity skeleton σ

PRM RU + entity skeleton σ

entity skeleton σ

PRM-RU + entity skeleton σ⇒ probability distribution over full instantiations I

Learning PRMs w/ RUIdea:

define scoring function do phased local search over legal structures

Key Components:Key Components:legal models

model new dependenciesscoring models

unchangedsearching model space

new operators

Legal Models

Review

Mood

Paper

Important

Paper

Important

Cites

Citing

Important

Accepted Accepted

CitingCited

Legal Models

Cites1.Selector

Cites1.CitedP2 Important

R1.Mood

P2.Important

P3 Important

P1.Accepted

P3.Important

P4.Importantp

When a node’s parent is defined using an uncertain

p

p grelation, the reference RV must be a parent of the node as well.

Structure Search

CitesCiting

PaperTopicW d

PaperTopicWords

PaperPaper

PaperPaperPaper

PaperPaper AuthorCiting

CitedWordsWordsp

PaperPaperPaper

PaperPaper

Institution

CitedPapers

1 01.0

Structure Search: New Operators

CitesCiting

PaperTopicW d

PaperTopicWords

PaperPaper

PaperPaperPaper

PaperP AuthorCiting

CitedWordsWords

Cited

PaperPaper

PaperPaper

PaperPaper

Institution

Cited

Topic PaperPaper

PaperPaper

PaperPaper

PaperPaperPaper

PaperPaper

Paper

Structure Search: New Operators

CitesCiting

PaperTopicW d

PaperTopicWords

PaperPaper

PaperPaperPaper

PaperPaper AuthorCiting

CitedWordsWords

Cited

pPaper

PaperPaper

PaperPaper

Institution

Cited

Topic Paper

PaperPaper

Paper

PaperPaper

PaperP Paper

PaperpPaper

Paper Institution

PaperPaper

PaperP PaperPaper

PaperPaperPaper

Paper

PRMs w/ RU SummaryDefine semantics for uncertainty over which entities are related to each otherSearch now includes operators Refine and Abstract for constructing foreign-key dependency modelProvides one simple mechanism for link uncertainty

Existence Uncertainty

? ??

Document CollectionDocument Collection

PRM w/ Exists Uncertainty

Paper

Cites

PaperTopicWords

PaperTopicW dWords Words

Exists

Dependency model for existence of relationship

Exists Uncertainty Example

Cites

PaperTopic

PaperTopicCitesWords Words

Exists

Citer.Topic Cited.Topic

0 995 0005Theory Theory

False True

0.995 0005Theory TheoryAITheory 0.999 0001

AIAI 0 993 0008

AI Theory 0.997 0003AIAI 0.993 0008

Introduce Exists RVs

Author #1 Author #2Author #1Area Inst Area Inst

Paper#2 Topic Paper#3Topic

WordN

Paper#1

Word1

Topic...

WordNWord1

... ... ...

Word1WordN

#1-#2Exists

#2-#3Exists

#2-#1Exists

#3-#1Exists

#1-#3Exists Exists

#3-#2

Introduce Exists RVs

Author #1 Author #2Author #1Area Inst Area Inst

Paper#2 Topic Paper#3Topic

WordN

Paper#1

Word1

Topic...

WordNWord1 WordNWord1

WordNWord1

... ... ...

Word1WordN Word1WordN

#1-#2Exists

#2-#3Exists

#2-#1Exists

#3-#1Exists

#1-#3Exists Exists ExistsExists Exists ExistsExists Exists

#3-#2

PRMs w/ EU Semantics

PaperP5

PaperP2 PaperPaper

P2Paper Paper P5Topic

AI

PaperP4Topic Theory


PaperP1

Topic ???

P5Topic

AI

PaperP4Topic Theory


PaperP1

Topic

???CitesExists

Paper

TopicWords

Paper

TopicWords

??? ???

object skeleton σPRM EU

Exists

PRM EU + object skeleton σ

j

PRM-EU + object skeleton σ⇒ probability distribution over full instantiations I

PRMs w/ EULearningIdea:



model new dependenciesscoring models

unchangedsearching model space

unchanged

PRMs with classesRelations organized in a class hierarchy

Venue

Subclasses inherit their probability model from superclasses

Journal Conference

Subclasses inherit their probability model from superclassesInstances are a special case of subclasses of size 1As you descend through the class hierarchy, you can have y g y yricher dependency models

e.g. cannot say Accepted(P1) <- Accepted(P2) (cyclic)but can say Accepted(JournalP1) <- Accepted(ConfP2)but can say Accepted(JournalP1) <- Accepted(ConfP2)

Type UncertaintyIs 1st-Venue a Journal or Conference ?Create 1st-Journal and 1st-Conference objectsCreate 1 Journal and 1 Conference objectsIntroduce Type(1st-Venue) variable with possible values Journal and ConferenceMake 1st-Venue equal to 1st-Journal or 1st-Conference according to value of Type(1st-Venue)

Learning PRM-CHs

Vote

TVProgramDatabase: PersonTVProgram

Instance I

Vote

R l ti l

TVProgram Person • Class hierarchy providedL l hi hRelational

Schema• Learn class hierarchy

Learning PRMs w/ CHIdea:



model new dependencies

scoring modelsunchanged

searching model spacenew operators

Guaranteeing Acyclicity w/ Subclasses

JournalJournalTopicQuality

C f P

PaperTopic

Accepted Conf-PaperTopicQuality

QualityAccepted

Accepted

Journal AcceptedJournal.AcceptedPaper.AcceptedPaper.Class

Conf-Paper.Accepted

Learning PRM-CHScenario 1: Class hierarchy is provided

New OperatorsS i li /I h itSpecialize/Inherit

AcceptedPaper

AcceptedJournal AcceptedConference AcceptedWorkshopAcceptedWorkshop

Learning Class HierarchyIssue: partially observable data setConstruct decision tree for class defined over attributes observed in training setNew operator

S lit l tt ib tSplit on class attributeRelated class attribute

Paper.Venue

conferenceworkshop

class1 class3

journal

class1 class2

high

Paper.Author.Fame

medium low

class4 class5 class6

PRMs w/ Class HierarchiesAllow us to:

Refine a “heterogenous” class into moreRefine a heterogenous class into more coherent subclassesRefine probabilistic model along classRefine probabilistic model along class hierarchy

Can specialize/inherit CPDsCan specialize/inherit CPDsConstruct new dependencies that were originally “acyclic”originally acyclic

Provides bridge from class-based model to instance based modelto instance-based model

Summary: Directed Frame-based ApproachesApproaches

Focus on objects and relationshipswhat types of objects are there and how are theywhat types of objects are there, and how are they related to each other?how does a property of an object depend on other properties (of the same or other objects)?

Representation supportRepresentation supportAttribute uncertainty Structural uncertaintyyClass Hierarchies

ff fEfficient Inference and Learning Algorithms

Markov Networks

A th 1 A th 2 F4F2 (F2 F4)

clique potential

Author1Fame

Author2Fame 0.6

f2 0 3

f4f2

f

F4F2 φ(F2,F4)

Author3Fame

Author4Fame

f2

f2

0.3

0.3

f4

f4

nodes = domain variablesedges = mutual influence

f4 1.5f2

parameters measuregcompatibility of values

Network structure encodes conditional independencies:I(A1 Fame, A4 Fame | A2 Fame, A3 Fame)

Markov Network Semantics

conditional l l f ll j i tF1 F2

conditionalindependenciesin MN structure

+localclique

potentials

full jointdistribution

over domain=F3 F4

)43()42()31()21(14321 ffffffff)fffP(f φφφφ )4,3()4,2()3,1()2,1(4321 34241312 ffffffffZ

)f,,ff,P(f φφφφ=

where Z is a normalizing factor that ensures that thewhere Z is a normalizing factor that ensures that theprobabilities sum to 1

G d li it t i tGood news: no acyclicity constraintsBad news: global normalization (1/Z)

Advantages of Undirected Models

Symmetric, non-causal interactionsyWeb: categories of linked pages are correlatedSocial nets: individual correlated with peersCannot introduce direct edges because

of cycles

Patterns involving multiple entitiesWeb: “triangle” patternsWeb: triangle patternsSocial nets: transitive relations

Relational Markov NetworksLocality:

Local probabilistic dependencies given by relational linksU i lUniversals:

Same dependencies hold for all objects linked in a particular pattern

Paper1Topic

Author1Area T2T1 φ(T1,T2)

Template potential

Venue

Topic1.8

AI 0.3

AIAI

TH

T2T1 φ(T1,T2)

Author2 Paper2

SubAreaAI

TH

TH

0.3

1.50.2

TH

AI

THAuthor2 Paper2TopicArea

TH 1.5TH

RMNso Semantics

• Instantiated RMN MNvariables: attributes of all objectsvariables: attributes of all objectsdependencies: determined by links & RMN

LearningDiscriminative trainingMax-margin Markov networksAssociative Markov networksAssociative Markov networks

Collective classification:Classifies multiple entities and links simultaneouslyExploits links & correlations between related entitiesentities

Collective Classification OverviewRelational

MarkovV Author

Training Data

NetworkVenue Author

Paper

LearningApprox.

Model Structure

g

New DataInference

ConclusionsApprox.

Inference

Example:

Train on one labeled conference/venuePredict labels on a new conference given papers and links

Maximum Likelihood

D

o1.x, o1.y*…

D

Estimation Classification…om.x, om.y*

maximizew

log P (D y* D x)

Estimation Classification

argmaxyP (D’ y | D’ x)f(x,y) log Pw(D.y*, D.x) Pw(D’.y | D’.x)

We don’t care about the jointdistribution P(D.X=x, D.Y=y*)

Maximum Conditional Likelihood

D

o1.x, o1.y*…

D

Estimation Classification…om.x, om.y*

maximizew

log P (D y* | D x)

Estimation Classification

argmaxyP (D’ y | D’ x)f(x,y) log Pw(D.y* | D.x) Pw(D’.y | D’.x)

[Lafferty et al ’01]

Learning RMNs

RegGrade

StudentIntelligence CC

Template potential φ

CourseDifficulty BA

BBBCCACB

Student2 Reg2GradeIntelligence

Difficulty

AAABAC

)},(...),(exp{),( 212121 yyfwyyfwyy CCCCAAAA ⋅++⋅=φ

)()f(ww D.XZD.Y,D.XD.XD.YP log)|(log −⋅=

)f()f( )|(ww wXDYDEXDYDXDYDP XDYDP .,..,.).|.(log ..−=∇

Learning RMNs

φMaximize L = log P(D.Grade,D.Intelligence|D.Difficulty)

φ(Reg1.Grade,Reg2.Grade)

CACBCC

CACBCC

CACBCC

Difficulty Intelligence

Gradelow / higheasy / hard ABC

Intelligence

Grade

Intelligence

Grade

AAABACBABBBCCA

AAABACBABBBCCA

AAABACBABBBCCADifficulty

I t lli

Intelligence

Grade

I t lli

Intelligence

Grade

I t lli

Intelligence

Grade

0 0.5 1 1.5 20 0.5 1 1.5 20 0.5 1 1.5 2Intelligence

Grade

Intelligence

Grade

Intelligence

Grade

Difficulty Intelligence

Grade ),( 21 ggfwL

DggAA

AA∑

∈

−=∂∂

Intelligence

Grade

Intelligence

Grade

).|,(),( 21,

21

,

21

21

DifficultyDggPggf

w

DggAA

Dgg

∑∈

∈∂

Markov Logic Networks[Richardson and Domingos 03,Singla & Domingos 05, Kok & Domingos 05]

A Markov Logic Network (MLN) is a set of pairs (F, w) whereF is a formula in first-order logicw is a real number

Together with a finite set of constants,it defines a Markov network with

One node for each grounding of each predicate in the MLNOne feature for each grounding of each formula F in the MLN, with the corresponding weight w

Example of an MLN

( ))()(),(,)()(_

)(_)(),(

ysmartxsmartyxauthorcoyxpacceptedpqualityhighx

pqualityhighxsmartpxauthorx

⇔⇒∀⇒∀

⇒∧∀

2.11.15.1

( )),(_),(),(,

)()(),(_,yxauthorcopyauthorpxauthorpyx

yyy⇒∧∃∀∞

Example of an MLN

( ))()(),(,)()(_

)(_)(),(



⇔⇒∀⇒∀

⇒∧∀

2.11.15.1

( )),(_),(),(,


yyy⇒∧∃∀∞

Suppose we have constants: alice, bob and p1

Example of an MLN

( ))()(),(,)()(_

)(_)(),(



⇔⇒∀⇒∀

⇒∧∀

2.11.15.1

( )),(_),(),(,


yyy⇒∧∃∀∞

Suppose we have constants: alice, bob and p1

co_author(bob,alice) co_author(alice,bob)

co author(alice alice) co author(bob bob)

smart(bob)smart(alice)th ( 1 b b)

co_author(alice,alice) co_author(bob,bob)

high quality(p1)


high_quality(p1)

accepted(p1)

Markov Logic Networks

Combine first-order logic and Markov networksSyntax: First-order logic + WeightsSemantics: Templates for Markov networks

I f KBMC M W lkS t MCMCInference: KBMC + MaxWalkSat + MCMCLearning: ILP + Pseudo-likelihood / discriminitive trainingtraining

Summary: Undirected Approaches

Focus on symmetric, non-causal relationshipsLike directed approaches, support collective classification


Rule-based Directed ModelsFrame-based Directed Models

Undirected ApproachesFrame-based Undirected ModelsRule-based Undirected Models

Themes: RepresentationBasic representational elements and focus

rules: factsframes: objectsprograms: processes

R ti d i t tRepresenting domain structurecontextrelational skeletonrelational skeletonnon-probabilistic language constructs

Representing local probabilitiesnoise factorsconditional probability tablesclique potentialsclique potentials

Themes: Representational IssuesStructural uncertainty

reference, exists, type, number, identityCombining probabilities from multiple sources

combining rulestiaggregation

Cyclicityruling out (stratification)ruling out (stratification)introducing timeguaranteed acyclic relationsundirected models

Functions and infinite chainsiterative approximationiterative approximation

Themes: InferenceInference on ground random variables

knowledge based model constructionInference at the first-order object level

first-order variable eliminationunification• unification

• quantification over populationsstructured variable eliminationmemoization

Utilizing entity-relations structureQuery directed inferenceQuery-directed inference

backward chaining on query and evidencelazy evaluationy

Themes: LearningLearning parameters

Parameter sharingg• rules apply many times• same type of object appears many times

same function is called many times• same function is called many times

Expectation-MaximizationLearning structureLearning structure

structure search legal modelsgscoring functionsearch operators

GoalsBy the end of this tutorial, hopefully, you will be:

1. able to distinguish among different SRL tasksg g2. able to represent a problem in one of several SRL

representations3. excited about SRL research problems and practical

applications

Many other interesting topics that I didn’t have timeMany other interesting topics that I didn t have time to cover…

ConclusionStatistical Relational Learning

Supports multi-relational, heterogeneous domainsSupports noisy, uncertain, non-IID datapp y, ,aka, real-world data!

Differences in approaches:rule-based vs frame-basedrule based vs. frame baseddirected vs. undirected

Many common issues:Need for collective classification and consolidationNeed for collective classification and consolidationNeed for aggregation and combining rulesNeed to handle labeled and unlabeled dataN d t h dl t t l t i tNeed to handle structural uncertaintyetc.

Great opportunity for combining rich logical representation d i f d l i ith hi hi l t ti ti land inference and learning with hierarchical statistical

models!!

statistical relational learning: a tutorial learning - university of

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