today’s topics 12/1/15cs 540 - fall 2015 (shavlik©), lecture 27, week 131 read chapter 21 (skip...

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Today’s Topics

12/1/15 CS 540 - Fall 2015 (Shavlik©), Lecture 27, Week 13

• Read Chapter 21 (skip Section 21.5) of textbook

• Exam THURSDAY Dec 17, 5:30-7:30pm (here)

• Review of Fall 2014 Final Dec 15

• TA Dmitry at Epic 5:30-7:30pm on Weds Dec 16?

• HW5 due Dec 8 (and no later than Dec 11)

• Probabilistic Logic

• Markov Logic Networks (MLNs)- a popular and successful probabilistic logic

• Collective Classification

CS 540 - Fall 2015 (Shavlik©), Lecture 27, Week 13

Logic & Probability: Two Major Math Underpinnings of AI

Logic

Probabilities

Add Probabilities

Add Relations

Statistical RelationalLearning

MLNs a popular approach

Slide 212/1/15

Statistical Relational Learning (Intro to SRL, Getoor & Tasker (eds), MIT Press, 2007)

• Pure Logic Too ‘Fragile’

everything must be either true or false

• Pure Statistics Doesn’t Capture/Accept General Knowledge Well (tell it once rather than label N ex’s)

x human(x) y motherOf(x, y)

• Many Approaches Created Over the Years, Especially Last Few

including some at UWisc

12/1/15 CS 540 - Fall 2015 (Shavlik©), Lecture 27, Week 13 3

Markov Logic Networks(Richards and Domingos, MLj, 2006)

• Use FOPC, but add weights to formulae (‘syntax’)

wgt=10

x,y,z motherOf(x, z) 𝝠 fatherOf(y, z) married(x, y)

- weights represent ‘penalty’ if a candidateworld state violates the rule

- for ‘pure’ logic, wgt = ∞

• Formulae interpreted (‘semantics’) as compact way to specify a type of graphical model called a Markov Net – like a Bayes net, but undirected arcs– probabilities in Markov nets specified by clique potentials,

but we won’t cover them in cs540


Pedro Domingos


Using an MLN (‘Inference’)

• Assume we have a large knowledge baseof probabilistic logic rules

• Assume we are given the truth values of N predicates (the ‘evidence’)

• We may be asked to estimate the most probable joint setting for M ‘query’ predicates

• Brute-force solution – Consider 2M possible ‘complete world states’– Calculate truth value of all grounded formula in each state– Return one with smallest total penalty for violated MLN rules

(or, equivalently, the one with largest sum of satisfied rules)Slide 512/1/15

Probability of Candidate World States

Prob(specific world state)

(1/Z) exp( weights of grounded formulae

that are true in this world state)

Z is a normalizing term; we need to sum over all possible

world states (challenging to estimate)

- A world state is a conjunction of predicates

(eg, married(John, Sue), …, friends(Bill, Ann) )

- if we only want the most probable world state,

we don’t need to compute Z

If a world state violates a rule with infinite weight,

probability of that world state is zero (why?)12/1/15 CS 540 - Fall 2015 (Shavlik©), Lecture 27, Week 13 6

Grounding the MLN Formulae (replacing variables with constants)

• Assume we have this domain knowledge

wgt = 2 x,y P(x,y) Q(x)

wgt = 3 x P(x,1) (R(1) R(x))

• And these constants: 1 and 2

• So we have these ‘grounded’ rules (wgts not shown):

P(1,1) Q(1) P(1,2) Q(1) P(2,1) Q(2) P(2,2) Q(2)

P(1,1) R(1) P(2,1) (R(1) R(2))

• Aside: Each grounded rule becomes a clique in Markov network

(like a CPT in a Bayes Net)


Simple MLN Example

Have: wgt=2 P Q wgt=7 P Q

Four possible world states

The normalizing term: Z = e2 + e9 + e7 + e9 e10

12/1/15 CS 540 - Fall 2015 (Shavlik©), Lecture 27, Week 13

P Q Probability (unnormalized) False False (1/Z) e2+0

False True (1/Z) e2+7

True False (1/Z) e0+7

True True (1/Z) e2+7

What is prob(P=true & Q=false)in ‘std’ logic?

8

1 / e8

1 / e

1 / e3

1 / e

Collective Classification

• Assume we need to predict the outputs for N examples

• Could knowing (probability) example i is true impact (probability) example j is true?

Ie, relaxing the iid assumption about examples

• For instance“If Alice and Bob are friends, then if Alice likes a movie, Bob (probably) does as well.”


Collective Classification in MLNs

• Imagine we have a bunch of inference rules for predicting likes(Person, Food)

• We could add this to our MLNIf livesIn(?Person1, ?City)

livesIn(?Person2, ?City) isaFood(?Food)

Then [ likes(?Person1, ?Food) likes(?Person2, ?Food) ] with wgt = 3

“People in the same city generally like the same sorts of food”


So if we predicted likes of N people, the MLN would be encouraged to give consistent answers


A Famous MLN Example(first rule modified to use ↔)

“Smoking frequently causes cancer.”

wgt = 3 x smokes(x) ↔ cancer(x) // Assume it’s the ONLY cause

“Friends of smokers are likely to smoke.”

wgt = 2 x,y friends(x, y) ˄ smokes(x) smokes(y)

Assume below are our facts and we want to know the probs of the four world states involving smoking or not of John and MaryA simple collective classification example (try yourself with THREE people!)

friends(Mary, Mary), friends(Mary, John), ¬ cancer(Mary)

friends(John, Mary), friends(John, John), cancer(John)

12/1/15 11

Don’t buy these →


A Famous MLN Example (2)

“Smoking frequently causes cancer.” GROUNDED

wgt = 3 smokes(J) ↔ cancer(J)

wgt = 3 smokes(M) ↔ cancer(M)

“Friends of smokers are likely to smoke.” GROUNDED

wgt = 2 friends(M, M) ˄ smokes(M) smokes(M)

wgt = 2 friends(M, J) ˄ smokes(M) smokes(J)

wgt = 2 friends(J, M) ˄ smokes(J) smokes(M)

wgt = 2 friends(J, J ) ˄ smokes(J) smokes(J)

FACTS

friends(M, M), friends(M, J) , ¬ cancer(M)

friends(J, M), friends(J, J), cancer(J)12/1/15 12



Possible Complete Word States

(1) friends(M,M), friends(M, J) , friends(J,M), friends(J, J)

¬ smokes(M), ¬ smokes(J), ¬ cancer(Mary), cancer(John)

(2) friends(M,M), friends(M, J), friends(J,M), friends(J, J)

¬ smokes(M), smokes(J), ¬ cancer(Mary), cancer(John)


smokes(M), ¬ smokes(J), ¬ cancer(Mary), cancer(John)


smokes(M), smokes(J), ¬ cancer(Mary), cancer(John)

12/1/15 13


A Famous MLN Example (4)Possible Complete Word States

(1) friends(M,M), friends(M,J), friends(J,M), friends(J,J)

¬ smokes(M), ¬ smokes(J), ¬ cancer(M), cancer(J)





wgt = 2 ¬ friends(M,M) ˅ ¬ smokes(M) ˅ smokes(M)

wgt = 2 ¬ friends(M,J) ˅ ¬ smokes(M) ˅ smokes(J)

wgt = 2 ¬ friends(J, M) ˅ ¬ smokes(J) ˅ smokes(M)

wgt = 2 ¬ friends(J,J ) ˅ ¬ smokes(J) ˅ smokes(J)

12/1/15 14

F

T

T

T

T

T

Sum of Wgts = 0 + 3 + 2 + 2 + 2 + 2 = 11



(2) friends(M,M), friends(M,J) , friends(J,M), friends(J,J)

¬ smokes(M), smokes(J), ¬ cancer(M), cancer(J)








wgt = 2 ¬ friends(J,J ˅ ¬ smokes(J) ˅ smokes(J)

12/1/15 15

T

T

T

T

F

T

Sum of Wgts = 3 + 3 + 2 + 2 + 0 + 2 = 12




smokes(M), ¬ smokes(J), ¬ cancer(M), cancer(J)








wgt = 2 ¬ friends(J,J) ˅ ¬ smokes(J) ˅ smokes(J)

12/1/15 16

F

F

T

F

T

T

Sum of Wgts = 0 + 0 + 2 + 0 + 2 + 2 = 6




smokes(M), smokes(J), ¬ cancer(M), cancer(J)








wgt = 2 ¬ friends(J,J) ˅ ¬ smokes(J) ˅ smokes(J)

12/1/15 17

T

F

T

T

T

T

Sum of Wgts = 3 + 0 + 2 + 2 + 2 + 2 = 11





¬ smokes(M), ¬ smokes(J), ¬ cancer(Mary), cancer(John) 11


¬ smokes(M), smokes(J), ¬ cancer(Mary), cancer(John) 12


smokes(M), ¬ smokes(J), ¬ cancer(Mary), cancer(John) 6


smokes(M), smokes(J), ¬ cancer(Mary), cancer(John) 11

12/1/15 18

Sum of Wgts of Satisfied

Rules




(1) friends(M,M), friends(M, J) , friends(J,M), friends(J,J)

¬ smokes(M), ¬ smokes(J), ¬ cancer(Mary), cancer(John) 11

(2) friends(M,M), friends(M, J) , friends(J,M), friends(J, J)

¬ smokes(M), smokes(J), ¬ cancer(Mary), cancer(John) 12


smokes(M), ¬ smokes(J), ¬ cancer(Mary), cancer(John) 6


smokes(M), smokes(J), ¬ cancer(Mary), cancer(John) 11

Z = e11 + e12 + e6 + e11 = e11 (1 + e + e-5 + 1) 4.7 e11

12/1/15 19

Sum of Wgts of Satisfied Rules

Prob = e11 / Z 1 / 4.7 = 0.21

Prob = e12 / Z e / 4.7 = 0.58

Prob = e 6 / Z (1 / 5) e-5 0.001

Prob = e11 / Z 1 / 4.7 = 0.21


A Famous MLN Example (9)(1) ¬ smokes(M), ¬ smokes(J), ¬ cancer(M), cancer(J)

lost 3 pts because John had cancer yet wasn’t a smoker

(2) ¬ smokes(M), smokes(J), ¬ cancer(M), cancer(J)

lost 2 pts because Friends J and M had diff smoking habits

(3) smokes(M), ¬ smokes(J), ¬ cancer(M), cancer(J)

lost 8 pts because of three reasons

(4) smokes(M), smokes(J), ¬ cancer(M), cancer(J)

lost 3 pts because Mary smoked but didn’t have cancer

If we had wgt=3 smoking cancer and wgt=2 cancer smoking

Then (1) and (4) would have scored differently (but slides already too crowded!)

If we had more people, we would have more clearly seen influence of collective classification - try it yourself!12/1/15 20


• Assume we know Prob( q(1,2) ) = 0.85

• We can represent this as

Prob( observedQ(1,2) ) = 1.0 // ie, absolute evidence

wgt = 2 observedQ(1,2) q(1,2)

Handling Probabilistic Evidence - What if the Given’s are Uncertain?

12/1/15 21

Grounded Networks can be Very LARGE!

Given

wgt=2 x,y,z Friends(x, y) Friends(y, z) Friends(x, z)

and a world with 109 people

How big is the grounded network?1018 nodes since we need all groundings of Friends(?X, ?Y)

(and the number of world states is 21018)

So SAMPLING methods needed

(and have been published)12/1/15 CS 540 - Fall 2015 (Shavlik©), Lecture 27, Week 13 22


Knowledge-Base Population(http://www.nist.gov/tac/2015/KBP/)

Given: Text Corpus (ie, ordinary English)

Do: Extract Facts about PeopleBorn(Person, Date)

AttendedCollege(Person, College, DateRange)

EmployedBy(Person, Company, DateRange)

SpouseOf(PersonA, PersonB, DateRange)

ParentOf(PersonA, PersonB, DateRange)

Died(Person, Date)

12/1/15 23

http://www.nist.gov/tac/2015/KBP/


Sample Advice for Collective Classification?

What might we say to an ML working on KBP?

Think about constraints across the relationsPeople are only married to one person at a time.

People usually have fewer than five children and rarely more than ten.

Typically one graduates from college in their 20’s.

Most people only have one job at a time.

One cannot go to college before they were born or after they died.

Almost always your children are born after you were.

People tend to marry people of about the same age.

People rarely live to be over 100 years & never over 125.

People don’t marry their children.… 12/1/15 24


Sample Advice for Collective Classification?

What might we say to an ML working on KBP?

Think about constraints across the relationsPeople are only married to one person at a time.

People usually have fewer than five children and rarely more than ten.

Typically one graduates from college in their 20’s.

Most people only have one job at a time.

One cannot go to college before they were born or after they died.

Almost always your children are born after you were.

People tend to marry people of about the same age.

People rarely live to be over 100 years & never over 125.

People don’t marry their children.… 12/1/15 25

When converted to MLN notation, these sentences of common-sense knowledge improve the results of information-extraction algorithms that simply extract each relation

independently (and noisily)


Scaling Up MLN Inference (see ICDM ‘12 paper by Niu et al. titled “Scaling Inference for Markov Logic via Dual Decomposition”)

We successfully ran in 1 day on the Knowledge Base Population task with

– 240 million facts (from 500 million web articles)– 64 billion logic sentences in the ground MLN– 5 terrabyte database (from GreenPlum, Inc)– 256 GB RAM, 40 cores on 4 machines– See `DeepDive/Wisci’ at

www.youtube.com/user/HazyResearch/videos

Slide 2612/1/15

http://ftp.cs.wisc.edu/machine-learning/shavlik-group/niu.icdm12.pdf

http://www.youtube.com/user/HazyResearch/videos

Learning MLNs

Like with Bayes Nets, need to learn

– Structure (ie, a rule set; could be given by user)

– Weights (can use gradient descent)

– There is a small literature on these tasks(some by my group)


MLN Challenges

• Estimating probabilities (‘inference’)can be cpu-intensive

usually need to use clever sampling methodssince # of world states is 0(2N)

• Interesting direction: lifted inference(reason at first-order level, rather than on grounded network)

• Structure learning and refinement is a major challenge


MLN Wrapup

• Appealing combo of first-order logicand prob/stats (the two primary math underpinnings of AI)

• Impressive results on real-world tasks

• Appealing approach to ‘knowledge refinement’1. Humans write (buggy) common-sense rules2. MLN algo learns weights (and maybe ‘edits’ rules)

• Computationally demanding (both learning MLNs and using them to answer queries)

• Other approaches to probabilistic logic exist; vibrant/exciting research area


today’s topics 12/1/15cs 540 - fall 2015 (shavlik©), lecture 27, week 131 read chapter 21 (skip...

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