Learning First-Order Probabilistic Models with Combining Rules Sriraam Natarajan Prasad Tadepalli...

Learning First-Order Learning First-Order Probabilistic Models Probabilistic Models

with Combining Ruleswith Combining RulesSriraam NatarajanSriraam NatarajanPrasad TadepalliPrasad TadepalliEric AltendorfEric Altendorf

Thomas G. DietterichThomas G. DietterichAlan FernAlan Fern

Angelo RestificarAngelo Restificar

School of EECSSchool of EECSOregon State UniversityOregon State University

First-order Probabilistic First-order Probabilistic ModelsModels

Combine the expressiveness of first-order logic Combine the expressiveness of first-order logic with the uncertainty modeling of the graphical with the uncertainty modeling of the graphical modelsmodels

Several formalisms already exist: Several formalisms already exist: Probabilistic Relational Models (PRMs)Probabilistic Relational Models (PRMs) Bayesian Logic Programs (BLPs)Bayesian Logic Programs (BLPs) Stochastic Logic Programs (SLPs)Stochastic Logic Programs (SLPs) Relational Bayesian Networks (RBNs)Relational Bayesian Networks (RBNs) Probabilistic Logic Programs (PLPs), …Probabilistic Logic Programs (PLPs), …

Parameter sharing and quantification allow Parameter sharing and quantification allow compact representationcompact representation

Multiple Parents Multiple Parents ProblemProblem

Often multiple objects are related to an object Often multiple objects are related to an object by the same relationshipby the same relationship One’s friend’s drinking habits influence one’s ownOne’s friend’s drinking habits influence one’s own A student’s GPA depends on the grades in the A student’s GPA depends on the grades in the

courses he takes courses he takes The size of a mosquito population depends on the The size of a mosquito population depends on the

temperature and the rainfall each day since the last temperature and the rainfall each day since the last freezefreeze

The target variable in each of these statements The target variable in each of these statements

has multiple influents (“parents” in Bayes net has multiple influents (“parents” in Bayes net jargon)jargon)

Population

Rain1Temp1 Rain2Temp2 Rain3Temp3

Multiple Parents for Multiple Parents for populationpopulation

■ Variable number of parents■ Large number of parents■ Need for compact parameterization

Solution 1: AggregatorsSolution 1: Aggregators

Population


AverageRainAverageTemp

Deterministic

Problem: Does not take into account the interaction between related parents Rain and Temp

Stochastic

Solution 2: Combining Solution 2: Combining RulesRules

Population


Population3Population1 Population20

10

20

30

40

1st Qtr 2nd Qtr 3r d Qtr 4th Qtr

0

10

20

30

1st Qtr 2nd Qtr 3r d Qtr 4th Qtr

0

20

40

60

80

1st Qtr 2nd Qtr 3rd Qtr 4th Qtr

0

20

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1s t Qtr 2nd Qtr 3r d Qtr 4th Qtr

• Top 3 distributions share parameters

• The 3 distributions are combined into one final distribution

First-order Conditional First-order Conditional Influence Language Influence Language

(FOCIL)(FOCIL) Task and role of a document influence its Task and role of a document influence its

folderfolder if {task(t), doc(d), role(d,r,t)} then r.id, t.id Qinf d.folderif {task(t), doc(d), role(d,r,t)} then r.id, t.id Qinf d.folder..

The folder of the source of the document The folder of the source of the document influences the folder of the document influences the folder of the document

if {doc(d1), doc(d2), source(d1,d2)} then d1.folder Qinf if {doc(d1), doc(d2), source(d1,d2)} then d1.folder Qinf d2.folderd2.folder

The difficulty of the course and the The difficulty of the course and the intelligence of the student influence intelligence of the student influence his/her GPAhis/her GPA

if (student(s), course(c), takes(s,c))} then s.IQ, c.difficulty Qinf if (student(s), course(c), takes(s,c))} then s.IQ, c.difficulty Qinf

s.gpas.gpa))

Combining Multiple Combining Multiple Instances of a Single Instances of a Single

StatementStatement

If {task(t), doc(d), role(d,r,t)} then If {task(t), doc(d), role(d,r,t)} then t.id, r.id Qinf (Mean) t.id, r.id Qinf (Mean) d.folderd.folder

t1.id

d.folder

d.folder d.folder

Mean

r1.id t2.id r2.id

A Different FOCIL A Different FOCIL Statement for the Same Statement for the Same

Target VariableTarget Variable

If {doc(s), doc(d), source(s,d) } then If {doc(s), doc(d), source(s,d) } then s.folder Qinf (Mean) s.folder Qinf (Mean) d.folderd.folder

d.folder

s2.folder

d.folder d.folder

Mean

s1.folder

Combining Multiple Combining Multiple StatementsStatements

Weighted Mean{Weighted Mean{

If {task(t), doc(d), role(d,r,t)} then If {task(t), doc(d), role(d,r,t)} then

t.id, r.id Qinf (Mean) d.foldert.id, r.id Qinf (Mean) d.folder

If {doc(s), doc(d), source(s,d) } then If {doc(s), doc(d), source(s,d) } then

s.folder Qinf (Mean) d.folders.folder Qinf (Mean) d.folder

}}

““Unrolled” Network for Unrolled” Network for Folder Prediction Folder Prediction

t1.id

d.folder d.folder

d.folder

Weighted Mean

d.folder d.folder

s2.folder

d.folder d.folder

Mean Mean

r1.id t2.id r2.ids1.folder

X11,1 X11,k…

1

X12,1 X12,k…

2

…

X1m1,k X1m1,k…

m1

Mean

X21,1 X21,k…

1

X22,1 X22,k…

2

…

X2m2,k X2m2,k…

m2

Mean

Weighted mean

Rule1 Rule2

Y

General Unrolled Network

Gradient Descent for Gradient Descent for Squared ErrorSquared Error

Squared errorSquared error

where

Gradient Descent for Gradient Descent for Loglikelihood Loglikelihood

LoglikelihoodLoglikelihood

, where

Learning the weightsLearning the weights

Mean Squared Mean Squared ErrorError

LoglikelihoodLoglikelihood

X11,1 X11,k…

1

…

X1m1,k X1m1,k…

m1

Mean

X21,1 X21,k…

1

…

X2m2,k X2m2,k…

m2

Mean

Weighted mean

w1w2

Y

Expectation-Maximization

11 1

m1 21 2

m2

1/m1 1/m1 1/m2 1/m2

EM learningEM learning Expectation-step: Compute the Expectation-step: Compute the

responsibilities of each instance of each ruleresponsibilities of each instance of each rule

Maximization-step: Compute the maximum Maximization-step: Compute the maximum likelihood parameters using responsibilities likelihood parameters using responsibilities as the countsas the counts

where n is the # of examples with 2 or more rules instantiated

Experimental SetupExperimental Setup

500 documents, 6 tasks, 2 roles, 11 folders500 documents, 6 tasks, 2 roles, 11 folders Each document typically has 1-2 task-role Each document typically has 1-2 task-role

pairspairs 25% of documents have a source folder25% of documents have a source folder 10-fold cross validation10-fold cross validation

Weighted Mean{Weighted Mean{

If {task(t), doc(d), role(d,r,t)} then t.id, r.id If {task(t), doc(d), role(d,r,t)} then t.id, r.id Qinf (Mean) d.folder.Qinf (Mean) d.folder.

If {doc(s), doc(d), source(s,d) } then If {doc(s), doc(d), source(s,d) } then s.folder Qinf (Mean) d.folder. }s.folder Qinf (Mean) d.folder. }

Folder prediction taskFolder prediction task

Mean reciprocal rank –Mean reciprocal rank –

where nwhere nii is the number of times the is the number of times the true folder was ranked as true folder was ranked as ii

Propositional classifiers:Propositional classifiers: Decision trees and Naïve BayesDecision trees and Naïve Bayes Features are the number of occurrences Features are the number of occurrences

of each task-role pair and source of each task-role pair and source document folderdocument folder

RankRank EMEM GD- GD- MSMS

GD-LLGD-LL J48J48 NBNB

11 349349 354354 346346 351351 326326

22 107107 9898 113113 100100 110110

33 2222 2626 1818 2828 3434

44 1515 1212 1515 66 1919

55 66 44 44 66 44

66 00 00 33 00 00

77 11 44 11 22 00

88 00 22 00 00 11

99 00 00 00 66 11

1010 00 00 00 00 00

1111 00 00 00 00 55

MRRMRR 0.8290.82999

0.8320.83255

0.8270.82744

0.8270.82799

0.7970.797

Lessons from Real-world Lessons from Real-world DataData

The propositional learners are almost as The propositional learners are almost as good as the first-order learners in this good as the first-order learners in this domain!domain! The number of parents is 1-2 in this domainThe number of parents is 1-2 in this domain About ¾ of the time only one rule is applicableAbout ¾ of the time only one rule is applicable Ranking of probabilities is easy in this caseRanking of probabilities is easy in this case

Accurate modeling of the probabilities is Accurate modeling of the probabilities is neededneeded Making predictions that combine with other Making predictions that combine with other

predictionspredictions Cost-sensitive decision makingCost-sensitive decision making

2 rules with 2 inputs each: W2 rules with 2 inputs each: Wrule1rule1= = 0.1,W0.1,Wrule2rule2= 0.9= 0.9

Probability that an example matches Probability that an example matches a rule = .5a rule = .5

If an example matches a rule, the If an example matches a rule, the number of instances is 3 - 10number of instances is 3 - 10

Performance metric: average Performance metric: average absolute error in predicted absolute error in predicted probabilityprobability

Synthetic Data SetSynthetic Data Set

Synthetic Data Set - Synthetic Data Set - ResultsResults

0

0.05

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0.15

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0.25

0.3

0.35

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Number of Training Examples

Avera

ge e

rro

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EMGDMSGDLLJ48NB

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Avera

ge E

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GDMS

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GDMS-Fixed

Synthetic Data Set GDMSSynthetic Data Set GDMS

Synthetic Data Set GDLLSynthetic Data Set GDLL

0

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Av

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Synthetic Data Set EMSynthetic Data Set EM

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Ave

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EM-True

EM-Fixed

ConclusionsConclusions

Introduced a general instance of multiple Introduced a general instance of multiple parents problem in first-order probabilistic parents problem in first-order probabilistic languageslanguages

Gradient descent and EM successfully Gradient descent and EM successfully learn the parameters of the conditional learn the parameters of the conditional distributions as well as the parameters of distributions as well as the parameters of the combining rules (weights)the combining rules (weights)

First order methods significantly First order methods significantly outperform propositional methods in outperform propositional methods in modeling the distributions when the modeling the distributions when the number of parents number of parents ¸̧ 3 3

Future WorkFuture Work

We plan to extend these results to We plan to extend these results to more general classes of combining more general classes of combining rulesrules

Develop efficient inference Develop efficient inference algorithms with combining rulesalgorithms with combining rules

Develop compelling applications Develop compelling applications Combining rules and aggregators Combining rules and aggregators

Can they both be understood as Can they both be understood as instances of causal independence?instances of causal independence?

Learning First-Order Probabilistic Models with Combining Rules Sriraam Natarajan Prasad Tadepalli...

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