an introduction to markov logic networks in knowledge bases

An Introduction to Markov Logic Networks

in Knowledge Bases9/30/2014

Sean [email protected]

From Big Data to Big Wisdom

From Big Data to Big Wisdom

Data Facts Rules

What is a Knowledge Base?

Graph Representation

Triple Representation

• FatherOf(Darth Vader, Luke Skywalker)• memberOf(Luke Skywalker, Rebel Alliance)• hasWeapon(Luke Skywalker, lightsaber)• isAlliedWith(Luke Skywalker, Han Solo)• IsAlliedWith(Luke Skywalker, Chewbacca)• livesOn(Luke Skywalker, Tatooine)• hasAdviser(Luke Skywalker, Yoda)• …..

Large Scale KBs

• KnowitAll (Univ. of Washington)• 500M facts extracted from 100M webpages (ClueWeb)

• DBPedia (Univ. of Leipzig)• 3.5M entities, 700M facts extracted from Wikipedia

• YAGO (Max-Planck Institute)• 2M entities, 20M facts extracted from Wikipedia and WordNet

• Freebase (Google)• 20M entities, 300M facts from integrated data sources and humans

• NELL (Carnegie Mellon)• 850M facts extracted from 500M webpages

Explicit vs. Implicit Information

Does Obama live in the US?

Knowledge Expansion

Inferring New Facts

• Markov Network• Undirected graphical model• Markov Property: Probabilities for each node depend only on its neighbors• Most probable assignment of random variables defined by cliques• HARD to figure out all the clique potentials

• First Order Logic• Data described by predicates with arguments• Logical formulas denote inference rules over predicates• Deterministic, ie. no probability• EASY to specify logical formulas

• Markov Logic: Markov Networks + First Order Logic• Statistical power of MLNs + specification power of FOL

Markov Networks

Cancer

CoughAsthma

Smoking

Smoking Cancer Ф(S,C)

False False 4.5

False True 4.5

True False 2.7

True True 4.5

c

cc xZxP )(

1)(

x c

cc xZ )(

Weight of Feature i Feature i

Markov Networks

Cancer

CoughAsthma

Smoking

iii xfw

ZxP )(exp

1)(

otherwise0

CancerSmokingif1)CancerSmoking,(1f

5.41 w

First Order Logic

Fact A Fact B

Fact A Fact B Fact C^

First Order Logic

isA(Luke, Jedi) memberOf(Luke, Rebel Alliance)

First Order Logic

isAlliedWith(Han, Luke)

memberOf(Luke, Rebel Alliance)^

memberOf(Han, Rebel Alliance)

First Order Logic

isAlliedWith(X, Y)

memberOf(Y, Z)^

memberOf(X, Z)

Soft Rules vs. Hard Rules

bornIn(X, Y) locatedIn(Y, Z)^ bornIn(X, Z)


bornIn(X, Y) locatedIn(Y, Z)^ bornIn(X, Z)HARD



friendOf(X, Y) friendOf(Y, Z)^ friendOf(X, Z)



friendOf(X, Y) friendOf(Y, Z)^ friendOf(X, Z)SOF

T



friendOf(X, Y) friendOf(Y, Z)^ friendOf(X, Z)SOF

T

MLN Basic Idea: Give every formula a weight (higher weight stronger constraint)

Markov Logic Networks: Logic + ProbabilityA Markov Logic Network is a collection of pairs (F, W)

memberOf(X, Z) ^ memberOf(Y, Z) -> alliedWith(X, Y) 8.5

hasLightsaber(X) -> memberOf(X, Rebels) 0.5

alliedWith(X, Y) -> alliedWith(Y, X) 100

Building a Markov Network

• Nodes = all possible instantiations of a predicate• Edge between two nodes iff they appear together in some formula


Han (H)Luke (L)Rebels (R)Empire (E)

memberOf(H, E)

memberOf(L, E)

alliedWith(H, L)



memberOf(H, E)

memberOf(L, E)

alliedWith(H, L)alliedWith(L, H)



memberOf(H, R)memberOf(L, R)

memberOf(H, E)

memberOf(L, E)





memberOf(H, E)

memberOf(L, E)






memberOf(H, E)

memberOf(L, E)


hasLightSaber(H)hasLightSaber(L)






memberOf(H, E)

memberOf(L, E)







c

cc xZxP )(

1)(


memberOf(H, E)

memberOf(L, E)







satisfiesit formulas of weightsexpP(world)

c

cc xZxP )(

1)(


memberOf(H, E)

memberOf(L, E)







Knowledge Base:

memberOf(Luke, Rebels)

memberOf(Han, Rebels)

hasLightSaber(Luke)

~memberOf(Luke, Empire)

Probability of Possible World

c

cc xZxP )(

1)(


)exp()( j

jjcc fwx

c

cc xZxP )(

1)(

otherwise0

X)(Y,alliedWithY)(X,alliedWithif1))X(Y,alliedWithY),(X,alliedWith(1f

1001 w

iii xnw

ZxP )(exp

1)(


Weight of formula i No. of true groundings of formula i in x

c

cc xZxP )(

1)(

)exp()( j

jjcc fwx

otherwise0

X)(Y,alliedWithY)(X,alliedWithif1))X(Y,alliedWithY),(X,alliedWith(1f

1001 w


memberOf(H, E)

memberOf(L, E)







iii xnw

ZxP )(exp

1)(

Inference in MLNs

• MAP Inference: Find the most likely possible world given some evidence

Inference in MLNs


i

iiy

yxnw ),(maxarg

Inference in MLNs


Solved with MaxWalkSAT

i

iiy

yxnw ),(maxarg

Inference in MLNs


Solved with MaxWalkSAT

• Marginal Inference: Find the marginal distribution of a predicate (node) given all the others

i

iiy

yxnw ),(maxarg

MaxWalkSAT

for i ← 1 to max-tries do solution = random truth assignment for j ← 1 to max-flips do if ∑ weights(sat. clauses) > threshold then return solution c ← random unsatisfied clause with probability p flip a random variable in c else flip variable in c that maximizes ∑ weights(sat. clauses) return failure, best solution found


memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)



memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)


1

1

0

1Fill in Observed Data


memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)


1

1

0

1?

?

?

?

Find Missing Data


memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)


1

1

0

10

0

0

0

Initialization Assignment


memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)


1

1

0

10

0

0

0

Flip One At Random


memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)


1

1

0

10

0

0

1

Flip One At Random


memberOf(H, E)

memberOf(L, E)







Knowledge Base:



hasLightSaber(Luke)


1

1

0

11

1

0

1

Flip One At Random

Learning in MLNs

• Data is a relational database or knowledge base

Knowledge Base:

memberOf(Luke, Rebels) hasLightSaber(Luke)

memberOf(Han, Rebels) ~memberOf(Luke, Empire)

Learning in MLNs


• Learning parameters (weights)• Similar to learning weights for Markov networks

Knowledge Base:



Learning in MLNs


• Learning parameters (weights)• Similar to learning weights for Markov networks

• Learning structure (formulas)• Inductive logic programming

Knowledge Base:



No. of times clause i is true in data

Expected no. times clause i is true according to MLN

Learning Weights via Max-Likelihood

)()()(log xnExnxPw iwiwi

iii xnw

ZxP )(exp

1)(

Alchemy (Open Source Software)

• Full first-order logic syntax• Inference (MAP and marginal probabilities)• Lifted Inference• Weight Learning (generative and discriminative)• Structure Learning• Large set of tutorials for learning

• Information extraction, social network modeling, entity resolution

alchemy.cs.washington.edu

Current Problem: Scale to Large KBs!• KnowitAll (Univ. of Washington)

• 500M facts extracted from 100M webpages (ClueWeb)

• DBPedia (Univ. of Leipzig)• 3.5M entities, 700M facts extracted from Wikipedia

• YAGO (Max-Planck Institute)• 2M entities, 20M facts extracted from Wikipedia and WordNet

• Freebase (Google)• 20M entities, 300M facts from integrated data sources and humans

• NELL (Carnegie Mellon)• 850M facts extracted from 500M webpages

an introduction to markov logic networks in knowledge bases

Documents

new knowledge

big data

washington500m facts

names of entities

chewbaccalivesonluke

explicit information

integrated data sources

automated scraping of