chapter 11: markov logic networks - machine learning

LEARNING AND INFERENCE IN GRAPHICAL MODELS

Chapter 11: Markov Logic Networks

Dr. Martin Lauer

University of FreiburgMachine Learning Lab

Karlsruhe Institute of TechnologyInstitute of Measurement and Control Systems

Learning and Inference in Graphical Models. Chapter 11 – p. 1/28

References for this chapter

◮ Matthew Richardson and Pedro Domingos, Markov logic networks. In:Machine Learning, vol. 62, no. 1-2, pp. 107-136, 2006


Logic and probabilities

Two perspectives on artificial intelligence:

Logical reasoning

◮ knowledge is stored in logicalformulas

◮ a logical formula is either true orfalse

◮ the world is sure

◮ the world is repeatable

◮ logical calcultio

Probabilistic reasoning

◮ knowledge is stored in jointprobability distributions

◮ we never know the exact value of arandom variable

◮ the world in unsure

◮ repetitions of an experiment yielddifferent results

◮ stochastic inference procedures

Can we combine logical and probabilistic reasoning?


Propositional logic

In propositional logic we have:

◮ Boolean variables a, b, . . .

◮ formulas created by combining Boolean variables (including true, false) withconnectors¬a, a ∧ b, a ∨ b, a→ b, a↔ b

◮ an interpretation is an assignment of Boolean variables to either true orfalse

◮ a model of a set of formulae is an interpretation for which all formulaeevaluate to true

Can we represent a set of propositional formulae in a MRF?


Markov logic networks

Let G = {g1, . . . , gn} be a set of Boolean functions over variables a1, . . . , ak

We create a factor graph with

◮ k binary variable nodes B1, . . . , Bk

◮ n factor nodes F1, . . . , Fn

◮ Bj is connected with Fi if Boolean variable aj occurs in formula gi

◮ we define potential functions for factor Fi as followsif Fi is connected to B

j(i)1, . . . , B

j(i)N

ψi(bj(i)1, . . . , b

j(i)Ni

) =

e1 if formula gi is satisfied for an interpretation

that maps al onto true if and only if bl = 1

e0 otherwise

The MRF that refers to this factor graph is called a Markov logic network (MLN)



For simplicity of notation, let us introduce

g̃i(bj(i)1, . . . , b

j(i)Ni

) :=

1 if formula gi is satisfied for an interpretation

that maps al onto true if and only if bl = 1

0 otherwise

Hence, potential functions can be written as

ψi(bj(i)1, . . . , b

j(i)Ni

) = eg̃i(b

j(i)1

,...,bj(i)Ni

)



Example:

g1 : a1 ∨ a2

g2 : a2 → a3

g3 : ¬a3

Potential functions:

ψ1(b1, b2) =

{

e if b1 + b2 ≥ 1

1 otherwise

ψ2(b2, b3) =

{

e if b3 − b2 ≥ 0

1 otherwise

ψ3(b3) =

{

e if b3 = 0

1 otherwise

B1

B2

B3

�F1 �F2 �F3



By construction of the Markov logic network,

◮ an interpretation of Boolean variables a1, . . . , ak induces values for randomvariables B1, . . . , Bk

◮ vice versa, values for random variables B1, . . . , Bk induce an interpretationof Boolean variables a1, . . . , ak

I(al) = true ⇔ Bl = 1

I(al) = false ⇔ Bl = 0



The joint probability of the Markov logic network:

p(b1, . . . , bk) =1

Z

n∏

i=1

ψi(bj(i)1, . . . , b

j(i)Ni

)

=1

Ze

∑ni=1 g̃i(b

j(i)1

,...,bj(i)Ni

)

The joint probability counts the number of satisfied formulae.

Conclusions:

1. If I is a model of the set of formulae, then the the induced values of Bi arethe maximum-a-posterior estimator.

2. If the set of formulae is satisfiable, then the maximum-a-posterior estimatorof the MLN induces a model.

3. If the set of formulae is unsatisfiable, then the maximum-a-posteriorestimator of the MLN induces an interpretation which satisfies as manyformulae as possible.


Complexity of finding the MAP

Remark:From the previous result we can conclude that determining the MAP estimator inan MRF is NP-hard.

Proof:We know that the satisfiability problem (SAT) is NP-complete.The construction of a MLN from a set of formulae can be done in polynomial timeHence, finding the MAP estimator in a MLN is NP-hard (even more, it isNP-complete).Hence, the complexity of finding the MAP estimator in a general MRF is NP-hard.



a1 a2 a3 b1 b2 b3 I(g1) I(g2) I(g3) p(·)

false false false 0 0 0 false true true 1Ze2

false false true 0 0 1 false true false 1Ze1

false true false 0 1 0 true false true 1Ze2

false true true 0 1 1 true true false 1Ze2

true false false 1 0 0 true true true 1Ze3

true false true 1 0 1 true true false 1Ze2

true true false 1 1 0 true false true 1Ze2

true true true 1 1 1 true true false 1Ze2

B1

B2

B3

�F1 �F2 �F3

g1 : a1 ∨ a2

g2 : a2 → a3

g3 : ¬a3



a1 a2 a3 b1 b2 b3 I(g1) I(g2) I(g3) I(g4) p(·)

false false false 0 0 0 false true true true 1Ze3

false false true 0 0 1 false true false false 1Ze1

false true false 0 1 0 true false true true 1Ze3

false true true 0 1 1 true true false false 1Ze2

true false false 1 0 0 true true true false 1Ze3

true false true 1 0 1 true true false false 1Ze2

true true false 1 1 0 true false true false 1Ze2

true true true 1 1 1 true true false false 1Ze2

B1

B2

B3

�F1 �F2 �F3

�

F4

g1 : a1 ∨ a2

g2 : a2 → a3

g3 : ¬a3

g4 : (a1 ↔ a3) ∧ ¬a1



The unsatisfiable case is most interesting since it is typical in real-worldapplications.

◮ we have a set of formulae which describe vague knowledge

◮ we want to infer which interpretation is “most likely”

◮ some formulae might be more important or less unsure than others



Example: Evening activity task.What to do this evening? General knowledge base:

g1 : cinema ∨ party

g2 : ¬(cinema ∧ party)

g3 : good film ↔ cinema

g4 : good people ↔ party

How should we decide if tonight good film and good people are both true?

good film

�good film ↔ cinema

cinema

good people

�good people ↔ party

party

�

cinema ∨ party

�

¬(cinema ∧ party)



Example: Evening activity task.Let us assume some of the rules are more important to us than others, e.g.

g1 : cinema ∨ party (not important, w1 = 1)

g2 : ¬(cinema ∧ party) (very important, cannot do both, w2 = 9)

g3 : good film ↔ cinema (not that important, w3 = 3)

g4 : good people ↔ party (important, w4 = 6)

Let us model importance of rules with non-negative weights.

Modify potential functions:

ψi(bj(i)1, . . . , b

j(i)Ni

) = ewig̃i(b

j(i)1

,...,bj(i)Ni

)

Now, the exponent in the joint probability sums the weights of satisfied formulae.

How should we decide if tonight good film and good people are both true?

→ go to party since MAP yields 1Ze16 (party) instead of 1

Ze13 (cinema)


Learning in Markov logic networks

How can we determine adequate weights wi?

In our evening activity example, assume we observe our behavior for a couple ofdays, i.e. for T days we know the truth value of cinema , party , good film , andgood people . Those observations serve as training examples.

Goal: find weights wi that maximize the log-likelihood for all training examples, i.e.

maximizew1,...,wn

T∑

t=1

log p(b(t)1 , . . . , b

(t)k |w1, . . . , wn)

with

log p(b(t)1 , . . . , b

(t)k |w1, . . . , wn) =

n∑

i=1

(wig̃i(b(t)

j(i)1

, . . . , b(t)

j(i)Ni

))−logZ(w1, . . . , wn)

Try maximization with gradient ascent approach.Difficulty:

calculating ∂∂wi

Z(w1, . . . , wn) requires exponential amount of time

→ approximative techniques, cf. Richardson and Domingos, 2006



Special case: we only have one logical fomula g1.

Goal: calculate the gradient of log p(b(t)1 , . . . , b

(t)k |w1)

log p(b(t)1 , . . . , b

(t)k |w1) = w1g̃1(b

(t)1 , . . . , b

(t)k )− logZ(w1)

with

Z1(w1) =∑

(b1,...,bk)∈{0,1}k

ewig̃i(b1,...,bk) = K+1 e

w1 +K−1

where

K+1 := |{b1, . . . , bk) ∈ {0, 1}k|g̃1(b1, . . . , bk) = 1}|

K−1 := |{b1, . . . , bk) ∈ {0, 1}k|g̃1(b1, . . . , bk) = 0}|



Hence,

log p(b(t)1 , . . . , b

(t)k |w1) = w1g̃1(b

(t)1 , . . . , b

(t)k )− logZ(w1))

= w1g̃1(b(t)1 , . . . , b

(t)k )− log(K+

1 ew1 +K−

1 ))

∂

∂w1

log p(b(t)1 , . . . , b

(t)k |w1) = g̃1(b

(t)1 , . . . , b

(t)k )−

K+1 e

w1

K+1 e

w1 +K−1

∂

∂w1

T∑

t=1

log p(b(t)1 , . . . , b

(t)k |w1) =

T∑

t=1

g̃1(b(t)1 , . . . , b

(t)k )

︸︷︷︸

=:G1

−T ·K+

1 ew1

K+1 e

w1 +K−1

G1 is the number of training examples in which g1 is satisfied



Zeroing the partial derivative:

0!=

∂

∂w1

T∑

t=1

log p(b(t)1 , . . . , b

(t)k |w1)

0 = G1 −T ·K+

1 ew1

K+1 e

w1 +K−1

· · ·

ew1 =GK−

1

(T −G)K+1

w1 = logGK−

1

(T −G)K+1

Observations:

◮ the more often g1 is satisfied in the training examples, the largerw1 becomes

◮ the more models g1 has, the smaller w1 becomes


Markov logic networks for propositional logic

Brief receipe of MLNs for propositional logic:

◮ creating and training a MLN

• create a set of propositional formulae that might contribute to a logicalinference

• collect training examples (the more the better)

• train the weights wi

◮ applying a MLN to a new situation

• determine all Boolean variables that you can observe

• apply inference procedures (e.g. Gibbs sampling) on the MLN to find theMAP estimator

• from the MAP estimator you can derive the best interpretation of thescene


First order logic

Can MLNs be generalized to first order logic (FOL)?

Formula in first order logic:

◮ variables x ∈ D

◮ constants c1, c2, . . .

◮ functions f : Dn → D

◮ predicates p : Dn → Bool

◮ logical constants true, false

◮ locical connectors ∧,∨,¬,→,↔

◮ quantifier ∀, ∃

Examples:

∀x p(x) ∨ ∃y (q(y, x) → p(y))

q(x, f(c, y)) ∧ ¬p(f(g(y), c))

. . .


First order logic

Yes, but we need some simplifications of FOL:

◮ finite set D

◮ there is exactly one constant for each element in D

◮ no functions

◮ no quantifiers

◮ all variables are implicitly ∀-quantified

What do we gain by these simplifications?→ a finite set of ground terms


Grounding

Definition: a ground term is a formula that does not contain variables.

We can create a ground term from any formula by substituting variables byconstants. We can create all ground terms of a formula by applying all possiblesubstitutions to the formula.

Example:assume the only constants c1, c2 and formula p(x, y) → q(x)

groundings:

p(c1, c1) → q(c1)

p(c1, c2) → q(c1)

p(c2, c1) → q(c2)

p(c2, c2) → q(c2)

Every ground formula can be interpreted as a formula of propositional logic. Now,we can apply the MLN idea to all ground formulae of a set of FOL formulae.


Markov logic networks for FOL

Remarks:

◮ grounding is a general concept for inference in FOL. Grounding itself doesnot restrict the expressive power of FOL

◮ in general, grounding yields an unbounded number of ground formulae evenfor simple examples

◮ our simplifications limit the number of ground formulae to a finite number

◮ but our simplifications also limit the expressive power of FOL to theexpressive power of propositional logic

◮ main reason for this reduction of expressive power: no functions, noexists-quantifiers, no Skolemization



Example:Maria says: “I don’t like boys. But I like all the people who my friends (=thepersons that I like) like. I like Susan and Susan likes Tom.

FOL formulae:

g1 : boy(x) → ¬like(Maria, x)

g2 : (like(Maria, x) ∧ like(x, y)) → like(Maria, y)

g3 : like(Maria, Susan)

g4 : like(Susan,Tom)

g5 : boy(Tom)

g6 : ¬boy(Maria)

g7 : ¬boy(Susan)



g1,1 : boy(Tom) → ¬like(Maria,Tom)

g1,2 : boy(Susan) → ¬like(Maria,Susan)

g1,3 : boy(Maria) → ¬like(Maria,Maria)

g2,1 : (like(Maria,Tom) ∧ like(Tom,Tom)) → like(Maria,Tom)

g2,2 : (like(Maria,Tom) ∧ like(Tom,Susan)) → like(Maria,Susan)

g2,3 : (like(Maria,Tom) ∧ like(Tom,Maria)) → like(Maria,Maria)

g2,4 : (like(Maria,Susan) ∧ like(Susan,Tom)) → like(Maria,Tom)

g2,5 : (like(Maria,Susan) ∧ like(Susan,Susan)) → like(Maria,Susan)

g2,6 : (like(Maria,Susan) ∧ like(Susan,Maria)) → like(Maria,Maria)

g2,7 : (like(Maria,Maria) ∧ like(Maria,Tom)) → like(Maria,Tom)

g2,8 : (like(Maria,Maria) ∧ like(Maria,Susan)) → like(Maria,Susan)

g2,9 : (like(Maria,Maria) ∧ like(Maria,Maria)) → like(Maria,Maria)

g3 : like(Maria,Susan)

g4 : like(Susan,Tom)

g5 : boy(Tom)

g6 : ¬boy(Maria)

g7 : ¬boy(Susan)

Ground formulae

Important:ground formulaethat are createdfrom the sameformula sharetheir weights wi



like(Maria,Maria)

like(Maria,Tom)

boy(Maria)boy(Tom) boy(Susan)

like(Tom,Susan) like(Maria,Susan) like(Susan,Susan)

like(Tom,Tom)

like(Tom,Maria) like(Susan,Maria)

like(Susan,Tom)

g7

g6

g5

g1,1 g

1,2g1,3

g2,3

g2,1

g2,2

g2,4

g2,5

g2,6

g2,7

g2,8

g2,9

g3

g4


Summary

◮ definition of Markov logic networks

◮ MAP estimators as logical models

◮ weighted formulae

◮ training a MAP from training examples

◮ extensions to first order logic


chapter 11: markov logic networks - machine learning

Documents