Probabilistic Soft Logic (PSL) is a declarative language for expressing collective probabilistic inference problems. PSL uses predicates, atoms, rules, and sets to define relationships and constraints between entities. A PSL program consists of a set of rules and an input database. PSL can be used for tasks like entity resolution and link prediction by expressing evidence through rules and performing probabilistic inference. Inference in PSL finds the most probable explanation for unknown atoms by minimizing a hinge-loss objective using an alternating direction method of multipliers algorithm. This scalable algorithm optimizes local copies of variables in parallel.

1. Probabilistic Soft Logic:
An Overview
Lise Getoor, Stephen Bach, Bert Huang
University of Maryland, College Park
http://psl.umiacs.umd.edu
Additional Contributors: Matthias Broecheler, Shobeir Fakhraei, Angelika Kimmig,
London, Alex Memory, Hui Miao, Lily Mihalkova, Eric Norris, Jay Pujara, Theo Rekatsinas, Arti Rames

2. Probabilistic Soft Logic (PSL)
Declarative language based on logics to express
collective probabilistic inference problems
 Predicate = relationship or property
 Atom = (continuous) random variable
 Rule = capture dependency or constraint
 Set = define aggregates
PSL Program = Rules + Input DB

3. Probabilistic Soft Logic (PSL)
Declarative language based on logics to express
collective probabilistic inference problems
 Predicate = relationship or property
 Atom = (continuous) random variable
 Rule = capture dependency or constraint
 Set = define aggregates
PSL Program = Rules + Input DB

4. Probabilistic Soft Logic
Entity Resolution
 Entities
- People References
 Attributes
- Name
 Relationships
- Friendship
 Goal: Identify
references that denote
the same person
A B
John Smith J. Smith
name name
C
E
D F G
H
friend friend
=
=

5. Probabilistic Soft Logic
Entity Resolution
 References, names,
friendships
 Use rules to express
evidence
- ‘’If two people have similar
names, they are probably the
same’’
- ‘’If two people have similar
friends, they are probably the
same’’
- ‘’If A=B and B=C, then A and C
must also denote the same
person’’
A B
John Smith J. Smith
name name
C
E
D F G
H
friend friend
=
=

6. Probabilistic Soft Logic
Entity Resolution
 References, names,
friendships
 Use rules to express
evidence
- ‘’If two people have similar
names, they are probably the
same’’
- ‘’If two people have similar
friends, they are probably the
same’’
- ‘’If A=B and B=C, then A and C
must also denote the same
person’’
A B
John Smith J. Smith
name name
C
E
D F G
H
friend friend
=
=
A.name ≈{str_sim} B.name => A≈B : 0.8

7. Probabilistic Soft Logic
Entity Resolution
 References, names,
friendships
 Use rules to express
evidence
- ‘’If two people have similar
names, they are probably the
same’’
- ‘’If two people have similar
friends, they are probably the
same’’
- ‘’If A=B and B=C, then A and C
must also denote the same
person’’
A B
John Smith J. Smith
name name
C
E
D F G
H
friend friend
=
=
{A.friends} ≈{} {B.friends} => A≈B : 0.6

8. Probabilistic Soft Logic
Entity Resolution
 References, names,
friendships
 Use rules to express
evidence
- ‘’If two people have similar
names, they are probably the
same’’
- ‘’If two people have similar
friends, they are probably the
same’’
- ‘’If A=B and B=C, then A and C
must also denote the same
person’’
A B
John Smith J. Smith
name name
C
E
D F G
H
friend friend
=
=
A≈B ^ B≈C => A≈C : ∞

9. Probabilistic Soft Logic
Link Prediction
 Entities
- People, Emails
 Attributes
- Words in emails
 Relationships
- communication, work
relationship
 Goal: Identify work
relationships
- Supervisor, subordinate,
colleague









10. Probabilistic Soft Logic
Link Prediction
 People, emails, words,
communication, relations
 Use rules to express
evidence
- “If email content suggests type X,
it is of type X”
- “If A sends deadline emails to B,
then A is the supervisor of B”
- “If A is the supervisor of B, and A
is the supervisor of C, then B and C
are colleagues”









11. Probabilistic Soft Logic
Link Prediction
 People, emails, words,
communication, relations
 Use rules to express
evidence
- “If email content suggests type X,
it is of type X”
- “If A sends deadline emails to B,
then A is the supervisor of B”
- “If A is the supervisor of B, and A
is the supervisor of C, then B and C
are colleagues”








complete by
due

12. Probabilistic Soft Logic
Link Prediction
 People, emails, words,
communication, relations
 Use rules to express
evidence
- “If email content suggests type X,
it is of type X”
- “If A sends deadline emails to B,
then A is the supervisor of B”
- “If A is the supervisor of B, and A
is the supervisor of C, then B and C
are colleagues”









13. Probabilistic Soft Logic
Link Prediction
 People, emails, words,
communication, relations
 Use rules to express
evidence
- “If email content suggests type X,
it is of type X”
- “If A sends deadline emails to B,
then A is the supervisor of B”
- “If A is the supervisor of B, and A
is the supervisor of C, then B and C
are colleagues”









14. Probabilistic Soft Logic

Node Labeling
?

15. Probabilistic Soft Logic

Voter Opinion Modeling
?
$$
Tweet
Status
update

16. Probabilistic Soft Logic

Voter Opinion Modeling

 
spouse
spouse
colleague
colleague
spouse
friend
friend
friend
friend

17. Probabilistic Soft Logic

Voter Opinion Modeling

 
vote(A,P) ∧ spouse(B,A)  vote(B,P) : 0.8
vote(A,P) ∧ friend(B,A)  vote(B,P) : 0.3
spouse
spouse
colleague
colleague
spouse
friend
friend
friend
friend

18. Probabilistic Soft Logic
Multiple Ontologies
18
Organization
Employees
Developer Staff
CustomersService & Products
Hardware IT Services Sales PersonSoftware
work forprovides
buys
helps
interacts
sells todevelops
Company
Employee
Technician Accountant
CustomerProducts & Services
Hardware Consulting SalesSoftware Dev
works fordevelop
buys
helps
interacts with
sells

19. Probabilistic Soft Logic
Ontology Alignment
19
Organization
Employees
Developer Staff
CustomersService & Products
Hardware IT Services Sales PersonSoftware
work forprovides
buys
helps
interacts
sells todevelops
Company
Employee
Technician Accountant
CustomerProducts & Services
Hardware Consulting SalesSoftware Dev
works fordevelop
buys
helps
interacts with
sells

20. Probabilistic Soft Logic
Ontology Alignment
20
Organization
Employees
Developer Staff
CustomersService & Products
Hardware IT Services Sales PersonSoftware
work forprovides
buys
helps
interacts
sells todevelops
Company
Employee
Technician Accountant
CustomerProducts & Services
Hardware Consulting SalesSoftware Dev
works fordevelop
buys
helps
interacts with
sells
Match, Don’t Match?

21. Probabilistic Soft Logic
Ontology Alignment
21
Organization
Employees
Developer Staff
CustomersService & Products
Hardware IT Services Sales PersonSoftware
work forprovides
buys
helps
interacts
sells todevelops
Company
Employee
Technician Accountant
CustomerProducts & Services
Hardware Consulting SalesSoftware Dev
works fordevelop
buys
helps
interacts with
sells
Similar to what extent?

22. Probabilistic Soft Logic
Logic Foundation

23. Probabilistic Soft Logic
Rules
 Atoms are real valued
- Interpretation I, atom A: I(A) [0,1]
- We will omit the interpretation and write A [0,1]
 ∨, ∧ are combination functions
- T-norms: [0,1]n →[0,1]
H1 ∨... Hm ← B1 ∧ B2 ∧ ... Bn
Ground Atoms
[Broecheler, et al., UAI ‘1

24. Probabilistic Soft Logic
Rules
 Combination functions (Lukasiewicz T-norm)
 A ∨ B = min(1, A + B)
 A ∧ B = max(0, A + B – 1)
H1 ∨... Hm ← B1 ∧ B2 ∧ ... Bn
[Broecheler, et al., UAI ‘1

25. Probabilistic Soft Logic
Satisfaction
 Establish Satisfaction
- ∨ (H1,..,Hm) ← ∧ (B1,..,Bn)
H1 ∨... Hm ← B1 ∧ B2 ∧ ... Bn
H1 ← B1:0.7 ∧ B2:0.8
[Broecheler, et al., UAI ‘1
≥0.5

26. Probabilistic Soft Logic
Distance to Satisfaction
 Distance to Satisfaction
- max( ∧ (B1,..,Bn) - ∨ (H1,..,Hm) , 0)
H1 ∨... Hm ← B1 ∧ B2 ∧ ... Bn
H1:0.7 ← B1:0.7 ∧ B2:0.8
H1:0.2 ← B1:0.7 ∧ B2:0.8
0.0
0.3
[Broecheler, et al., UAI ‘1

27. Probabilistic Soft Logic
W: H1 ∨... Hm ← B1 ∧ B2 ∧ ... Bn
Rule Weights
 Weighted Distance to Satisfaction
- d(R,I) = W  max( ∧ (B1,..,Bn) - ∨ (H1,..,Hm) , 0)
[Broecheler, et al., UAI ‘1

28. Probabilistic Soft Logic
Let’s Review
 Given a data set and a PSL program, we can
construct a set of ground rules.
 Some of the atoms have fixed truth values
and some have unknown truth values.
 For every assignment of truth values to the
unknown atoms, we get a set of weighted
distances from satisfaction.
 How to decide which is best?

29. Probabilistic Soft Logic
Probabilistic Foundation

30. Probabilistic Soft Logic
Probabilistic Model
Probability
density over
interpretation I
Normalization
constant
Set of ground
rules
Distance
exponent
in {1, 2}
Rule’s weight Rule’s distance to satisfaction

31. Probabilistic Soft Logic
Hinge-loss Markov Random Fields
Subject to arbitrary linear constraints
PSL models ground out to HL-MRFs
Log-concave!

32. Probabilistic Soft Logic
MPE Inference

33. Probabilistic Soft Logic
Inferring Most Probable Explanations
Objective:
Convex optimization
Decomposition-based inference
algorithm using the ADMM framework

34. Probabilistic Soft Logic
Alternating Direction Method of
Multipliers
• We perform inference using the alternating
direction method of multipliers (ADMM)
• Inference with ADMM
is fast, scalable,
and straightforward
• Optimize subproblems
(ground rules) independently, in parallel
• Auxiliary variables enforce consensus
across subproblems
[Bach et al., NIPS ’12]

35. Probabilistic Soft Logic
Inference Algorithm
Initialize local copies
of variables and
Lagrange multipliers
Begin inference
iterations
Simple updates
solve subproblems
for each potential...
...and each
constraint
Average to update
global variables and
clip to [0,1]
[Bach, et al., Under revie

36. Probabilistic Soft Logic
ADMM Scalability
0
100
200
300
400
500
600
125000 225000 325000
Timeinseconds
Number of potential functions and constraints
ADMM
Interior-point method
[Bach, et al., NIPS ‘12]

37. Probabilistic Soft Logic
ADMM Scalability [Bach, et al., Under revie

38. Probabilistic Soft Logic
Distributed MPE Inference:
MapReduce Implementation
z1z1 zq zqz1 z2z2 zp
sub
problem
local
variable
copy
Mapper
z1 z2z1 z1 z2 zq zq zp
Reducer
update
global
component
load global variable
X as side data
Job Bootstrap
HDFS or
HBase
read/write
subproblem
write new
global variable
read global
variable X
Hui Miao, Xiangyang Liu

39. Probabilistic Soft Logic
Distributed MPE Inference:
GraphLab Implementation
.
.
.
.
.
.
.
.
.
.
.
.
sub
problem
node
global
variable
component
gather
get z
apply
update y
update x
scatter
notify z
gather
get local z,y
apply
update z
scatter
unless converge
notify X
update i update i+1
Hui Miao, Xiangyang Liu

40. Probabilistic Soft Logic
Weight Learning

41. Probabilistic Soft Logic
Weight Learning
Learn from training data
No need to hand-code rule-weights
Various methods:
- approximate maximum likelihood
[Broecheler et al., UAI ’10]
- maximum pseudolikelihood
- large-margin estimation
[Bach, et al., UAI ’13]

42. Probabilistic Soft Logic
Weight Learning
State-of-the-art supervised-learning
performance on
- Collective classification
- Social-trust prediction
- Preference prediction
- Image reconstruction
[Bach, et al., UAI ’13]

43. Probabilistic Soft Logic
Foundations Summary

44. Probabilistic Soft Logic
Foundations Summary
Design probabilistic models using
declarative language
- Syntax based on first-order logic
Inference of most-probable
explanation is fast convex
optimization (ADMM)
Learning algorithms for training rule
weights from labeled data

45. Probabilistic Soft Logic
PSL Applications

46. Probabilistic Soft Logic
Document Classification




2
2
2
2
2A
B
A or B?
A or B?
A or B?
 Given a networked collection of documents
 Observe some labels
 Predict remaining labels using
 link direction
 inferred class label
[Bach, et al., UAI ’13]

47. Probabilistic Soft Logic
Computer Vision Applications
Low-level vision:
- image reconstruction
High-level vision:
- activity recognition in videos

48. Probabilistic Soft Logic
Image Reconstruction
RMSE reconstruction error
[Bach, et al., UAI ’13]

49. Probabilistic Soft Logic
Activity Recognition in Videos
crossing waiting queueing walking talking dancing jogging
[London, et al., Under revi

50. Probabilistic Soft Logic
Results on Activity Recognition
Recall matrix between
different activity types
Accuracy metrics
compared against
baseline features
[London, et al., Under revi

51. Probabilistic Soft Logic
Social Trust Prediction
Competing models from social
psychology of strong ties
- Structural balance [Granovetter ’73]
- Social status [Cosmides et al., ’92]
Leskovec, Huttenlocher, & Kleinberg,
[2010]: effects of both models present
in online social networks

52. Probabilistic Soft Logic
Structural Balance vs. Social Status
 Structural balance: strong ties are governed by
tendency toward balanced triads
- e.g., the enemy of my enemy...
 Social status: strong ties indicate unidirectional
respect, “looking up to”, expertise status
- e.g., patient-nurse-doctor, advisor-advisee

53. Probabilistic Soft Logic
Structural Balance in PSL
[Huang, et al., SBP ‘1

54. Probabilistic Soft Logic
Structural Balance in PSL
[Huang, et al., SBP ‘1

55. Probabilistic Soft Logic
Social Status in PSL
[Huang, et al., SBP ‘1

56. Probabilistic Soft Logic
Social Status in PSL
[Huang, et al., SBP ‘1

57. Probabilistic Soft Logic
Experiments
 User-user trust ratings from two different
online social networks
 Observe some ratings, predict held-out
 Eight-fold cross validation on two data sets:
- FilmTrust - movie review network,
trust ratings from 1-10
- Epinions - product review network,
trust / distrust ratings {-1, 1}
[Huang, et al., SBP ‘1

58. Probabilistic Soft Logic
Compared Methods
 TidalTrust: graph-based propagation of trust
- Predict trust via breadth-first search to combine
closest known relationships
 EigenTrust: spectral method for trust
- Predict trustworthiness of nodes based on
eigenvalue centrality of weighted trust network
 Average baseline: predict average trust score for
all relationships
[Huang, et al., SBP ‘1

59. Probabilistic Soft Logic
FilmTrust Experiment
 Normalize [1,10] rating to [0,1]
 Prune network to largest connected-component
 1,754 users, 2,055 relationships
 Compare mean average error, Spearman’s rank
coefficient, and Kendall-tau distance
* measured on only non-default predictions
• PSL-Status and PSL-Balance disagree on 514 relationships
[Huang, et al., SBP ‘1

60. Probabilistic Soft Logic
Epinions Experiment
 Snowball sample of 2,000 users from
Epinions data set
 8,675 trust scores normalized to {0,1}
 Measure area under precision-recall
curve for distrust edges (rarer class)
[Huang, et al., SBP ‘1

61. Probabilistic Soft Logic
Knowledge Graph Identification
 Problem: Collectively reason about noisy,
inter-related fact extractions
 Task: NELL fact-promotion (web-scale IE)
- Millions of extractions, with entity ambiguity
and confidence scores
- Rich ontology: Domain, Range, Inverse,
Mutex, Subsumption
 Goal: Determine which facts to include in
NELL’s knowledge base
Jay Pujara, Hui Miao, William Cohen

62. Probabilistic Soft Logic
PSL Rules
Candidate Extractions:
- CandRel_CMC(E1,E2,R) => Rel(E1,E2,R)
- CandLbl_Morph(E1,L) => Lbl(E1,L)
Entity Resolution:
- SameEntity(E1,E2) & Lbl(E1,L) => Lbl(E2,L)
- SameEntity(E1,E2) & Rel(E1,X,R) => Rel(E2,X,R)
Ontological Rules:
- Domain(R,L) & Rel(E1,E2,R) => Lbl(E1,L)
- Mut(L1,L2) & Lbl(E,L1) => ~Lbl(E,L2)

63. Probabilistic Soft Logic
Results
Data: Iteration 165 of NELL
- 1.2M extractions
- 68K ontological constraints
- 14K labeled instances (Jiang, 2012)
- NELL & MLN baselines (Jiang, 2012)
F1 Precision Recall AUC-PR
NELL .673 .801 .580 .765
MLN .836 .837 .836 .899
PSL .841 .773 .922 .882

64. Probabilistic Soft Logic
Schema Matching
 Correspondences between
source and target schemas
 Matching rules
- ‘’If two concepts are the same,
they should have similar
subconcepts’’
- ‘’If the domains of two attributes
are similar, they may be the
same’’
Organization
Customers
Service &
Products
provides
buys
Company
Customer
Products &
Services
develop
buys
Portfolios
includes
develop(A, B) <= provides(A, B)
Company(A) <= Organization(A)
Products&Services(B) <= Service&Products(B)

65. Probabilistic Soft Logic
Schema Mapping
 Input: Schema matches
 Output: S-T query pairs
(TGD) for exchange or
mediation
 Mapping rules
- “Every matched attribute should
participate in some TGD.”
- “The solutions to the queries in
TGDs should be similar.”
Organization
Customers
Service &
Products
provides
buys
Company
Customer
Products &
Services
develop
buys
Portfolios
includes
∃Portfolio P, develop(A, P) ∧
includes(P, B) <= provides(A, B) . . .
Alex Memory, Angelika Kimmig

66. Probabilistic Soft Logic
Learning Latent Groups
 Can we better understand political discourse in social
media by learning groups of similar people?
 Case study: 2012 Venezuelan Presidential Election
 Incumbent: Hugo Chávez
 Challenger: Henrique Capriles
Left: This photograph was produced by Agência Brasil, a public Brazilian news agency. This file is licensed under the Creative
Commons Attribution 3.0 Brazil license. Right: This photograph was produced by Wilfredor. This file is licensed under the Creative
Commons Attribution-Share Alike 3.0 Unported license.
[Bach, et al., Under revi

67. Probabilistic Soft Logic
Learning Latent Groups
 South American tweets collected from 48-hour
window around election.
 Selected 20 top users
 Candidates, campaigns, media, and most
retweeted
 1,678 regular users interacted with at least one
top user and used at least one hashtag in another
tweet
 Those regular users had 8,784 interactions with
non-top users
[Bach, et al., Under revi

68. Probabilistic Soft Logic
Learning Latent Groups
[Bach, et al., Under revi

69. Probabilistic Soft Logic
Learning Latent Groups
[Bach, et al., Under revi

70. Probabilistic Soft Logic
Learning Latent Groups
[Bach, et al., Under revi

71. Probabilistic Soft Logic
Learning Latent Groups
[Bach, et al., Under revi

72. Probabilistic Soft Logic
Other LINQS Projects
 Drug-target interaction: Shobeir Fakhraei
 MOOC user-role prediction and expert identification: Arti Ramesh
 Semantic-role labeling: Arti Ramesh and Alex Memory
 Optimal source selection for data integration: Theo Rekatsinas
 Learning theory in non-I.I.D. settings: Ben London, Bert Huang
Collective stability [London et al, ICML13]
 Scalable graph analysis using GraphLab: Hui Miao, Walaa Eldin
Moustafa
Uncertain graph models: Walaa Eldin Moustafa

73. Probabilistic Soft Logic
Conclusion

74. Probabilistic Soft Logic
Conclusion
 PSL:
 expressive framework for structured problems
 scalable
 Much ongoing work, including incorporating hidden
variables and structure learning, distributing
 Very interested in applying it to a variety of
domains, especially in computational social science
 We encourage you to try it!
http://psl.umiacs.umd.edu

75. Probabilistic Soft Logic
psl.umiacs.umd.edu