open-world probabilistic databases - csweb.cs.ucla.edu/~guyvdb/slides/sum16.pdf · open-world...
TRANSCRIPT
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Open-World
Probabilistic Databases
Guy Van den Broeck
Scalable Uncertainty Management (SUM)
Sep 21, 2016
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Overview
1. Why probabilistic databases?
2. How probabilistic query evaluation?
3. Why open world?
4. How open-world query evaluation?
5. What is the broader picture?
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Why probabilistic databases?
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What we’d like to do…
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What we’d like to do…
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Google Knowledge Graph
> 570 million entities > 18 billion tuples
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• Tuple-independent probabilistic database
• Learned from the web, large text corpora, ontologies,
etc., using statistical machine learning.
Co
au
tho
r
Probabilistic Databases
x y P
Erdos Renyi 0.6
Einstein Pauli 0.7
Obama Erdos 0.1
Scie
nti
st x P
Erdos 0.9
Einstein 0.8
Pauli 0.6
[Suciu’11]
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Information Extraction
x y P
Luc Laura 0.7
Luc Hendrik 0.6
Luc Kathleen 0.3
Luc Paol 0.3
Luc Paolo 0.1
Coauthor
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Information Extraction
x y P
Luc Laura 0.7
Luc Hendrik 0.6
Luc Kathleen 0.3
Luc Paol 0.3
Luc Paolo 0.1
Coauthor
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Noisy!
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Noisy!
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• Relational data is increasingly probabilistic
– NELL machine reading (>50M tuples)
– Google Knowledge Vault (>2BN tuples)
– DeepDive (>7M tuples)
• Next step: Probabilistic Query Evaluation SQL or First-order logic
Q(x) = ∃y Scientist(x)∧Coauthor(x,y)
SELECT Scientist.X FROM Scientist, Coauthor WHERE Scientist.X = Coauthor.Y
[Carlson’10, Dong’14, Niu’12+
Probabilistic Databases
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What we’d like to do…
∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
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Einstein is in the Knowledge Graph
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Erdős is in the Knowledge Graph
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This guy is in the Knowledge Graph
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This guy is in the Knowledge Graph
… and he published with both Einstein and Erdos!
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Desired Query Answer
Ernst Straus
Barack Obama, …
Justin Bieber, …
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Desired Query Answer
Ernst Straus
Barack Obama, …
Justin Bieber, …
1. Fuse uncertain
information from web
⇒ Embrace probability!
2. Cannot come from
labeled data
⇒ Embrace query eval!
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[Chen’16+ (NYTimes)
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How probabilistic
query evaluation?
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Tuple-Independent Probabilistic DB
x y P
A B p1
A C p2
B C p3
Probabilistic database D:
Co
auth
or
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x y
A B
A C
B C
Tuple-Independent Probabilistic DB
x y P
A B p1
A C p2
B C p3
Possible worlds semantics:
p1p2p3
Probabilistic database D:
Co
auth
or
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x y
A B
A C
B C
Tuple-Independent Probabilistic DB
x y P
A B p1
A C p2
B C p3
Possible worlds semantics:
p1p2p3
(1-p1)p2p3
Probabilistic database D:
x y
A C
B C C
oau
tho
r
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x y
A B
A C
B C
Tuple-Independent Probabilistic DB
x y P
A B p1
A C p2
B C p3
Possible worlds semantics:
p1p2p3
(1-p1)p2p3
(1-p1)(1-p2)(1-p3)
Probabilistic database D:
x y
A C
B C
x y
A B
A C
x y
A B
B C
x y
A B x y
A C x y
B C x y
Co
auth
or
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x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) =
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
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x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) = 1-(1-q1)*(1-q2)
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
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x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) = 1-(1-q1)*(1-q2) p1*[ ]
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
![Page 29: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/29.jpg)
x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) = 1-(1-q1)*(1-q2) p1*[ ]
1-(1-q3)*(1-q4)*(1-q5)
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
![Page 30: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/30.jpg)
x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) = 1-(1-q1)*(1-q2) p1*[ ]
1-(1-q3)*(1-q4)*(1-q5) p2*[ ]
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
![Page 31: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/31.jpg)
x y P
A D q1 Y1
A E q2 Y2
B F q3 Y3
B G q4 Y4
B H q5 Y5
x P
A p1 X1
B p2 X2
C p3 X3
P(Q) = 1-(1-q1)*(1-q2) p1*[ ]
1-(1-q3)*(1-q4)*(1-q5) p2*[ ]
1- {1- } *
{1- }
Probabilistic Query Evaluation
Q = ∃x∃y Scientist(x) ∧ Coauthor(x,y)
Scie
nti
st
Co
auth
or
![Page 32: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/32.jpg)
Lifted Inference Rules
Preprocess Q (omitted), Then apply rules (some have preconditions)
*Suciu’11+
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Lifted Inference Rules
Preprocess Q (omitted), Then apply rules (some have preconditions)
P(¬Q) = 1 – P(Q) Negation
*Suciu’11+
![Page 34: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/34.jpg)
Lifted Inference Rules
P(Q1 ∧ Q2) = P(Q1) P(Q2) P(Q1 ∨ Q2) =1 – (1– P(Q1)) (1–P(Q2))
Preprocess Q (omitted), Then apply rules (some have preconditions)
Decomposable ∧,∨
P(¬Q) = 1 – P(Q) Negation
*Suciu’11+
![Page 35: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/35.jpg)
Lifted Inference Rules
P(Q1 ∧ Q2) = P(Q1) P(Q2) P(Q1 ∨ Q2) =1 – (1– P(Q1)) (1–P(Q2))
P(∃z Q) = 1 – ΠA ∈Domain (1 – P(Q[A/z])) P(∀z Q) = ΠA ∈Domain P(Q[A/z])
Preprocess Q (omitted), Then apply rules (some have preconditions)
Decomposable ∧,∨
Decomposable ∃,∀
P(¬Q) = 1 – P(Q) Negation
*Suciu’11+
![Page 36: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/36.jpg)
Lifted Inference Rules
P(Q1 ∧ Q2) = P(Q1) P(Q2) P(Q1 ∨ Q2) =1 – (1– P(Q1)) (1–P(Q2))
P(∃z Q) = 1 – ΠA ∈Domain (1 – P(Q[A/z])) P(∀z Q) = ΠA ∈Domain P(Q[A/z])
P(Q1 ∧ Q2) = P(Q1) + P(Q2) - P(Q1 ∨ Q2) P(Q1 ∨ Q2) = P(Q1) + P(Q2) - P(Q1 ∧ Q2)
Preprocess Q (omitted), Then apply rules (some have preconditions)
Decomposable ∧,∨
Decomposable ∃,∀
Inclusion/ exclusion
P(¬Q) = 1 – P(Q) Negation
*Suciu’11+
![Page 37: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/37.jpg)
Limitations
H0 = ∀x∀y Smoker(x) ∨ Friend(x,y) ∨ Jogger(y)
The decomposable ∀-rule:
P(∀z Q) = ΠA ∈Domain P(Q[A/z])
[Suciu’11]
![Page 38: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/38.jpg)
Limitations
H0 = ∀x∀y Smoker(x) ∨ Friend(x,y) ∨ Jogger(y)
The decomposable ∀-rule:
… does not apply:
H0[Alice/x] and H0[Bob/x] are dependent:
∀y (Smoker(Alice) ∨ Friend(Alice,y) ∨ Jogger(y))
∀y (Smoker(Bob) ∨ Friend(Bob,y) ∨ Jogger(y))
Dependent
P(∀z Q) = ΠA ∈Domain P(Q[A/z])
[Suciu’11]
![Page 39: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/39.jpg)
Limitations
H0 = ∀x∀y Smoker(x) ∨ Friend(x,y) ∨ Jogger(y)
The decomposable ∀-rule:
… does not apply:
H0[Alice/x] and H0[Bob/x] are dependent:
∀y (Smoker(Alice) ∨ Friend(Alice,y) ∨ Jogger(y))
∀y (Smoker(Bob) ∨ Friend(Bob,y) ∨ Jogger(y))
Dependent
Lifted inference sometimes fails.
Computing P(H0) is #P-hard in size database
P(∀z Q) = ΠA ∈Domain P(Q[A/z])
[Suciu’11]
![Page 40: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/40.jpg)
Are the Lifted Rules Complete?
You already know:
• Inference rules: PTIME data complexity
• Some queries: #P-hard data complexity
[Dalvi and Suciu;JACM’11]
![Page 41: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/41.jpg)
Are the Lifted Rules Complete?
You already know:
• Inference rules: PTIME data complexity
• Some queries: #P-hard data complexity
Dichotomy Theorem for UCQ / Mon. CNF
• If lifted rules succeed, then PTIME query
• If lifted rules fail, then query is #P-hard
[Dalvi and Suciu;JACM’11]
![Page 42: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/42.jpg)
Are the Lifted Rules Complete?
You already know:
• Inference rules: PTIME data complexity
• Some queries: #P-hard data complexity
Dichotomy Theorem for UCQ / Mon. CNF
• If lifted rules succeed, then PTIME query
• If lifted rules fail, then query is #P-hard
Lifted rules are complete for UCQ!
[Dalvi and Suciu;JACM’11]
![Page 43: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/43.jpg)
Why open world?
![Page 44: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/44.jpg)
Knowledge Base Completion
Given:
Learn:
Complete:
0.8::Coauthor(x,y) :- Coauthor(x,z) ∧ Coauthor(z,y).
x y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
… … …
x y P
Straus Pauli 0.504
… … …
Co
au
tho
r
![Page 45: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/45.jpg)
Bayesian Learning Loop
Bayesian view on learning:
1. Prior belief:
P(Coauthor(Straus,Pauli)) = 0.01
2. Observe page
P(Coauthor(Straus,Pauli| ) = 0.2
3. Observe page
P(Coauthor(Straus,Pauli)| , ) = 0.3
Principled and sound reasoning!
![Page 46: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/46.jpg)
Problem: Broken Learning Loop
Bayesian view on learning:
1. Prior belief:
P(Coauthor(Straus,Pauli)) = 0
2. Observe page
P(Coauthor(Straus,Pauli| ) = 0.2
3. Observe page
P(Coauthor(Straus,Pauli)| , ) = 0.3
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 47: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/47.jpg)
Problem: Broken Learning Loop
Bayesian view on learning:
1. Prior belief:
P(Coauthor(Straus,Pauli)) = 0
2. Observe page
P(Coauthor(Straus,Pauli| ) = 0.2
3. Observe page
P(Coauthor(Straus,Pauli)| , ) = 0.3
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 48: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/48.jpg)
Problem: Broken Learning Loop
Bayesian view on learning:
1. Prior belief:
P(Coauthor(Straus,Pauli)) = 0
2. Observe page
P(Coauthor(Straus,Pauli| ) = 0.2
3. Observe page
P(Coauthor(Straus,Pauli)| , ) = 0.3
[Ceylan, Darwiche, Van den Broeck; KR’16]
This is mathematical nonsense!
![Page 49: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/49.jpg)
What we’d like to do…
∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Ernst Straus
Kristian Kersting, …
Justin Bieber, …
![Page 50: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/50.jpg)
Open World DB
• What if fact missing?
• Probability 0 for:
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
![Page 51: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/51.jpg)
Open World DB
• What if fact missing?
• Probability 0 for:
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
![Page 52: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/52.jpg)
Open World DB
• What if fact missing?
• Probability 0 for:
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
![Page 53: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/53.jpg)
Open World DB
• What if fact missing?
• Probability 0 for:
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
![Page 54: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/54.jpg)
Open World DB
• What if fact missing?
• Probability 0 for:
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
Q5 = Coauthor(Einstein,Bieber) ∧ ¬Coauthor(Einstein,Bieber)
![Page 55: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/55.jpg)
Intuition
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … … Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 56: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/56.jpg)
Intuition
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
We know for sure that P(Q1) ≥ P(Q3), P(Q1) ≥ P(Q4)
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 57: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/57.jpg)
Intuition
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
We know for sure that P(Q1) ≥ P(Q3), P(Q1) ≥ P(Q4)
and P(Q3) ≥ P(Q5), P(Q4) ≥ P(Q5)
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
Q5 = Coauthor(Einstein,Bieber) ∧ ¬Coauthor(Einstein,Bieber)
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 58: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/58.jpg)
Intuition
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
We know for sure that P(Q1) ≥ P(Q3), P(Q1) ≥ P(Q4)
and P(Q3) ≥ P(Q5), P(Q4) ≥ P(Q5) because P(Q5) = 0.
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
Q5 = Coauthor(Einstein,Bieber) ∧ ¬Coauthor(Einstein,Bieber)
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 59: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/59.jpg)
Intuition
X Y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
We know for sure that P(Q1) ≥ P(Q3), P(Q1) ≥ P(Q4)
and P(Q3) ≥ P(Q5), P(Q4) ≥ P(Q5) because P(Q5) = 0.
We have strong evidence that P(Q1) ≥ P(Q2).
Q1 = ∃x Coauthor(Einstein,x) ∧ Coauthor(Erdos,x)
Q2 = ∃x Coauthor(Bieber,x) ∧ Coauthor(Erdos,x)
Q3 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
Q4 = Coauthor(Einstein,Bieber) ∧ Coauthor(Erdos,Bieber)
Q5 = Coauthor(Einstein,Bieber) ∧ ¬Coauthor(Einstein,Bieber)
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 60: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/60.jpg)
Problem: Curse of Superlinearity
• Reality is worse!
• Tuples are
intentionally
missing!
• Every tuple
has 99%
probability
![Page 61: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/61.jpg)
“This is all true, Guy,
but it’s just a temporary issue.”
“No
it’s not!
• A single table (Sibling)
• Facebook scale (billions of people)
• Real (non-zero) Bayesian beliefs
x y P
… … …
Sibling ⇒ 200 Exabytes of data”
Problem: Curse of Superlinearity
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 62: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/62.jpg)
Problem: Curse of Superlinearity
All Google storage is
a couple exabytes…
![Page 63: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/63.jpg)
Problem: Curse of Superlinearity
We should
be here!
![Page 64: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/64.jpg)
Problem: Evaluation
Given:
Learn:
0.8::Coauthor(x,y) :- Coauthor(x,z) ∧ Coauthor(z,y).
x y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
… … …
Co
au
tho
r
0.6::Coauthor(x,y) :- Affiliation(x,z) ∧ Affiliation(y,z).
OR
[De Raedt et al; IJCAI’15]
![Page 65: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/65.jpg)
Problem: Evaluation
Given:
Learn:
0.8::Coauthor(x,y) :- Coauthor(x,z) ∧ Coauthor(z,y).
x y P
Einstein Straus 0.7
Erdos Straus 0.6
Einstein Pauli 0.9
… … …
Co
au
tho
r
0.6::Coauthor(x,y) :- Affiliation(x,z) ∧ Affiliation(y,z).
OR
What is the likelihood, precision, accuracy, …?
[De Raedt et al; IJCAI’15]
![Page 66: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/66.jpg)
Open-World Prob. Databases
Intuition: tuples can be added with P < λ
Q2 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
P(Q2) ≥ 0
![Page 67: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/67.jpg)
Open-World Prob. Databases
Intuition: tuples can be added with P < λ
Q2 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Erdos Straus λ
Coauthor
P(Q2) ≥ 0
![Page 68: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/68.jpg)
Open-World Prob. Databases
Intuition: tuples can be added with P < λ
Q2 = Coauthor(Einstein,Straus) ∧ Coauthor(Erdos,Straus)
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Coauthor
X Y P
Einstein Straus 0.7
Einstein Pauli 0.9
Erdos Renyi 0.7
Kersting Natarajan 0.8
Luc Paol 0.1
… … …
Erdos Straus λ
Coauthor
0.7 * λ ≥ P(Q2) ≥ 0
![Page 69: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/69.jpg)
Closed-World Prob. Databases
![Page 70: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/70.jpg)
Open-World Prob. Databases
[Ceylan, Darwiche, Van den Broeck; KR’16]
![Page 71: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/71.jpg)
How open-world query
evaluation?
![Page 72: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/72.jpg)
UCQ / Monotone CNF
• Lower bound = closed-world probability
• Upper bound = probability after adding all
tuples with probability λ
![Page 73: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/73.jpg)
UCQ / Monotone CNF
• Lower bound = closed-world probability
• Upper bound = probability after adding all
tuples with probability λ
• Polynomial time☺
![Page 74: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/74.jpg)
UCQ / Monotone CNF
• Lower bound = closed-world probability
• Upper bound = probability after adding all
tuples with probability λ
• Polynomial time☺
• Quadratic blow-up
• 200 exabytes … again
![Page 75: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/75.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
![Page 76: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/76.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
Decomposable ∀-Rule
![Page 77: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/77.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
Decomposable ∀-Rule
Check independence:
Scientist(A) ∧ ∃y Coauthor(A,y)
Scientist(B) ∧ ∃y Coauthor(B,y)
![Page 78: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/78.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
Decomposable ∀-Rule
Check independence:
Scientist(A) ∧ ∃y Coauthor(A,y)
Scientist(B) ∧ ∃y Coauthor(B,y)
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 79: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/79.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
Decomposable ∀-Rule
Check independence:
Scientist(A) ∧ ∃y Coauthor(A,y)
Scientist(B) ∧ ∃y Coauthor(B,y)
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
Complexity PTIME
![Page 80: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/80.jpg)
Closed-World Lifted Query Eval
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 81: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/81.jpg)
Closed-World Lifted Query Eval
No supporting facts
in database!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 82: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/82.jpg)
Closed-World Lifted Query Eval
No supporting facts
in database!
Probability 0 in closed world
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 83: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/83.jpg)
Closed-World Lifted Query Eval
No supporting facts
in database!
Probability 0 in closed world
Ignore these queries!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 84: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/84.jpg)
Closed-World Lifted Query Eval
No supporting facts
in database!
Complexity linear time!
Probability 0 in closed world
Ignore these queries!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 85: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/85.jpg)
Open-World Lifted Query Eval
No supporting facts
in database!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 86: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/86.jpg)
Open-World Lifted Query Eval
No supporting facts
in database!
Probability p in closed world
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 87: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/87.jpg)
Open-World Lifted Query Eval
No supporting facts
in database!
Complexity PTIME!
Probability p in closed world
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 88: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/88.jpg)
Open-World Lifted Query Eval
No supporting facts
in database!
Probability p in closed world
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 89: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/89.jpg)
Open-World Lifted Query Eval
No supporting facts
in database!
Probability p in closed world
All together, probability (1-p)k
Do symmetric lifted inference
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 90: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/90.jpg)
Open-World Lifted Query Eval
No supporting facts
in database!
Probability p in closed world
All together, probability (1-p)k
Do symmetric lifted inference Complexity linear time!
Q = ∃x ∃y Scientist(x) ∧ Coauthor(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 91: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/91.jpg)
[Ceylan’16]
Complexity Results
![Page 92: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/92.jpg)
What is the broader picture?
![Page 93: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/93.jpg)
...
A Simple Reasoning Problem
?
Probability that Card1 is Hearts?
[Van den Broeck; AAAI-KRR’15]
![Page 94: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/94.jpg)
...
A Simple Reasoning Problem
?
Probability that Card1 is Hearts? 1/4
[Van den Broeck; AAAI-KRR’15]
![Page 95: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/95.jpg)
A Simple Reasoning Problem
...
?
Probability that Card52 is Spades given that Card1 is QH?
[Van den Broeck; AAAI-KRR’15]
![Page 96: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/96.jpg)
A Simple Reasoning Problem
...
?
Probability that Card52 is Spades given that Card1 is QH? 13/51
[Van den Broeck; AAAI-KRR’15]
![Page 97: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/97.jpg)
Let us automate this: 1. Probabilistic graphical model (e.g., factor graph)
2. Probabilistic inference algorithm (e.g., variable elimination or junction tree)
Automated Reasoning
[Van den Broeck; AAAI-KRR’15+
![Page 98: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/98.jpg)
Classical Reasoning
A
B C
D E
F
A
B C
D E
F
A
B C
D E
F
Tree Sparse Graph Dense Graph
• Higher treewidth • Fewer conditional independencies • Slower inference
![Page 99: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/99.jpg)
Let us automate this: 1. Probabilistic graphical model (e.g., factor graph)
is fully connected!
2. Probabilistic inference algorithm (e.g., variable elimination or junction tree) builds a table with 5252 rows
Automated Reasoning
(artist's impression)
[Van den Broeck; AAAI-KRR’15+
![Page 100: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/100.jpg)
Lifted Inference in SRL
Statistical relational model (e.g., MLN)
As a probabilistic graphical model: 26 pages; 728 variables;
676 factors 1000 pages; 1,002,000 variables;
1,000,000 factors
Highly intractable? – Lifted inference in milliseconds!
3.14 FacultyPage(x) ∧ Linked(x,y) ⇒ CoursePage(y)
![Page 101: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/101.jpg)
...
Tractable Reasoning
What's going on here?
Which property makes reasoning tractable?
[Niepert and Van den Broeck, AAAI’14], [Van den Broeck, AAAI-KRR’15]
![Page 102: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/102.jpg)
...
Tractable Reasoning
What's going on here?
Which property makes reasoning tractable?
⇒ Lifted Inference
High-level (first-order) reasoning
Symmetry
Exchangeability
[Niepert and Van den Broeck, AAAI’14], [Van den Broeck, AAAI-KRR’15]
![Page 103: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/103.jpg)
Model Counting
• Model = solution to a propositional logic formula Δ
• Model counting = #SAT
Rain Cloudy Model?
T T Yes
T F No
F T Yes
F F Yes
#SAT = 3
+
Δ = (Rain ⇒ Cloudy)
![Page 104: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/104.jpg)
First-Order Model Counting
Model = solution to
first-order logic
formula Δ
Δ = ∀d (Rain(d)
⇒ Cloudy(d))
Days = {Monday}
![Page 105: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/105.jpg)
First-Order Model Counting
Model = solution to
first-order logic
formula Δ
Δ = ∀d (Rain(d)
⇒ Cloudy(d))
Days = {Monday}
Rain(M) Cloudy(M) Model?
T T Yes
T F No
F T Yes
F F Yes
FOMC = 3
+
![Page 106: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/106.jpg)
First-Order Model Counting
Model = solution to
first-order logic
formula Δ
Δ = ∀d (Rain(d)
⇒ Cloudy(d))
Days = {Monday
Tuesday}
![Page 107: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/107.jpg)
First-Order Model Counting
Model = solution to
first-order logic
formula Δ
Rain(M) Cloudy(M) Rain(T) Cloudy(T) Model?
T T T T Yes
T F T T No
F T T T Yes
F F T T Yes
T T T F No
T F T F No
F T T F No
F F T F No
T T F T Yes
T F F T No
F T F T Yes
F F F T Yes
T T F F Yes
T F F F No
F T F F Yes
F F F F Yes
#SAT = 9
+
Δ = ∀d (Rain(d)
⇒ Cloudy(d))
Days = {Monday
Tuesday}
![Page 108: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/108.jpg)
FOMC Inference
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 109: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/109.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 110: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/110.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 111: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/111.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 112: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/112.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 113: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/113.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 114: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/114.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 115: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/115.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 116: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/116.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 117: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/117.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 118: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/118.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
→ models
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 119: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/119.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
If we know that there are k smokers?
→ models
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 120: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/120.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
If we know that there are k smokers?
→ models
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
→ models
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 121: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/121.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
If we know that there are k smokers?
In total…
→ models
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
→ models
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 122: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/122.jpg)
FOMC Inference
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
If we know that there are k smokers?
In total…
→ models
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
→ models
→ models
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 123: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/123.jpg)
Let us automate this:
Relational model
Lifted probabilistic inference algorithm
∀p, ∃c, Card(p,c)
∀c, ∃p, Card(p,c)
∀p, ∀c, ∀c’, Card(p,c) ∧ Card(p,c’) ⇒ c = c’
...
[Van den Broeck; AAAI-KRR’15]
![Page 124: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/124.jpg)
...
Playing Cards Revisited
∀p, ∃c, Card(p,c) ∀c, ∃p, Card(p,c)
∀p, ∀c, ∀c’, Card(p,c) ∧ Card(p,c’) ⇒ c = c’
[Van den Broeck.; AAAI-KR’15]
![Page 125: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/125.jpg)
...
Playing Cards Revisited
∀p, ∃c, Card(p,c) ∀c, ∃p, Card(p,c)
∀p, ∀c, ∀c’, Card(p,c) ∧ Card(p,c’) ⇒ c = c’
[Van den Broeck.; AAAI-KR’15]
![Page 126: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/126.jpg)
...
Playing Cards Revisited
∀p, ∃c, Card(p,c) ∀c, ∃p, Card(p,c)
∀p, ∀c, ∀c’, Card(p,c) ∧ Card(p,c’) ⇒ c = c’
Computed in time polynomial in n
[Van den Broeck.; AAAI-KR’15]
![Page 127: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/127.jpg)
Open-World Lifted Query Eval
All together, probability (1-p)k
Q = ∃x ∃y Smoker(x) ∧ Friend(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 128: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/128.jpg)
Open-World Lifted Query Eval
All together, probability (1-p)k
Open-world query evaluation on empty db
= Symmetric lifted inference
Q = ∃x ∃y Smoker(x) ∧ Friend(x,y)
P(Q) = 1 - ΠA ∈ Domain (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
= 1 - (1 - P(Scientist(A) ∧ ∃y Coauthor(A,y))
x (1 - P(Scientist(B) ∧ ∃y Coauthor(B,y))
x (1 - P(Scientist(C) ∧ ∃y Coauthor(C,y))
x (1 - P(Scientist(D) ∧ ∃y Coauthor(D,y))
x (1 - P(Scientist(E) ∧ ∃y Coauthor(E,y))
x (1 - P(Scientist(F) ∧ ∃y Coauthor(F,y))
…
![Page 129: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/129.jpg)
Even on #P-hard queries!
If we know precisely who smokes, and there are k smokers?
k
n-k
k
n-k
If we know that there are k smokers?
In total…
→ models
Database: Smokes(Alice) = 1 Smokes(Bob) = 0 Smokes(Charlie) = 0 Smokes(Dave) = 1 Smokes(Eve) = 0 ...
→ models
→ models
Smokes Smokes Friends
Δ = ∀x,y, (Smokes(x) ∧ Friends(x,y) ⇒ Smokes(y)) Domain = {n people}
![Page 130: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/130.jpg)
Tractable Classes
FO2
CNF
FO2
Safe monotone CNF Safe type-1 CNF
FO3
CQs
[VdB; NIPS’11+, [VdB et al.; KR’14], [Gribkoff, VdB, Suciu; UAI’15+, [Beame, VdB, Gribkoff, Suciu; PODS’15+, etc.
![Page 131: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/131.jpg)
Tractable Classes
FO2
CNF
FO2
Safe monotone CNF Safe type-1 CNF
FO3
CQs
[VdB; NIPS’11+, [VdB et al.; KR’14], [Gribkoff, VdB, Suciu; UAI’15+, [Beame, VdB, Gribkoff, Suciu; PODS’15+, etc.
![Page 132: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/132.jpg)
Tractable Classes
FO2
CNF
FO2
Safe monotone CNF Safe type-1 CNF
?
FO3
CQs
Δ = ∀x,y,z, Friends(x,y) ∧ Friends(y,z) ⇒ Friends(x,z)
[VdB; NIPS’11+, [VdB et al.; KR’14], [Gribkoff, VdB, Suciu; UAI’15+, [Beame, VdB, Gribkoff, Suciu; PODS’15+, etc.
![Page 133: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/133.jpg)
X Y
Smokes(x)
Gender(x)
Young(x)
Tall(x)
Smokes(y)
Gender(y)
Young(y)
Tall(y)
Properties Properties
FO2 is liftable!
![Page 134: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/134.jpg)
X Y
Smokes(x)
Gender(x)
Young(x)
Tall(x)
Smokes(y)
Gender(y)
Young(y)
Tall(y)
Properties Properties
Friends(x,y)
Colleagues(x,y)
Family(x,y)
Classmates(x,y)
Relations
FO2 is liftable!
![Page 135: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/135.jpg)
X Y
Smokes(x)
Gender(x)
Young(x)
Tall(x)
Smokes(y)
Gender(y)
Young(y)
Tall(y)
Properties Properties
Friends(x,y)
Colleagues(x,y)
Family(x,y)
Classmates(x,y)
Relations
FO2 is liftable!
“Smokers are more likely to be friends with other smokers.” “Colleagues of the same age are more likely to be friends.”
“People are either family or friends, but never both.” “If X is family of Y, then Y is also family of X.”
“If X is a parent of Y, then Y cannot be a parent of X.”
![Page 136: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/136.jpg)
Uncertainty in AI
Probability Distribution
=
Qualitative
+
Quantitative
![Page 137: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/137.jpg)
Probabilistic Graphical Models
Probability Distribution
=
Graph Structure
+
Parameterization
![Page 138: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/138.jpg)
Probabilistic Graphical Models
Probability Distribution
=
Graph Structure
+
Parameterization
+
![Page 139: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/139.jpg)
Weighted Model Counting
Probability Distribution
=
SAT Formula
+
Weights
[Chavira et al. 2008, Sang et al. 2005]
![Page 140: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/140.jpg)
Weighted Model Counting
Probability Distribution
=
SAT Formula
+
Weights
+
Rain ⇒ Cloudy
Sun ∧ Rain ⇒ Rainbow
w( Rain)=1
w(¬Rain)=2
w( Cloudy)=3
w(¬Cloudy)=5
…
[Chavira et al. 2008, Sang et al. 2005]
![Page 141: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/141.jpg)
Weighted First-Order
Model Counting
Probability Distribution
=
First-Order Logic
+
Weights
[Van den Broeck 2011, 2013, Gogate 2011]
![Page 142: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/142.jpg)
Weighted First-Order
Model Counting
Probability Distribution
=
First-Order Logic
+
Weights
+
Smokes(x) ∧ Friends(x,y)
⇒ Smokes(y)
w( Smokes(a))=1
w(¬Smokes(a))=2
w( Smokes(b))=1
w(¬Smokes(b))=2
w( Friends(a,b))=3
w(¬Friends(a,b))=5
…
[Van den Broeck 2011, 2013, Gogate 2011]
![Page 143: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/143.jpg)
Generalized Model Counting
Probability Distribution
=
Logic
+
Weights
![Page 144: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/144.jpg)
Generalized Model Counting
Probability Distribution
=
Logic
+
Weights
+
Logical Syntax
Model-theoretic
Semantics
Weight function w(.)
![Page 145: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/145.jpg)
Weighted Model Integration
Probability Distribution
=
SMT(LRA)
+
Weights
[Belle et al. IJCAI’15, UAI’15]
![Page 146: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/146.jpg)
Weighted Model Integration
Probability Distribution
=
SMT(LRA)
+
Weights
+
0 ≤ height ≤ 200
0 ≤ weight ≤ 200
0 ≤ age ≤ 100
age < 1 ⇒
height+weight ≤ 90
w(height))=height-10
w(¬height)=3*height2
w(¬weight)=5
…
[Belle et al. IJCAI’15, UAI’15]
![Page 147: Open-World Probabilistic Databases - CSweb.cs.ucla.edu/~guyvdb/slides/SUM16.pdf · Open-World Probabilistic Databases Guy Van den Broeck Scalable Uncertainty Management (SUM) Sep](https://reader034.vdocuments.us/reader034/viewer/2022042223/5eca5aeb7ecca31bce530a22/html5/thumbnails/147.jpg)
Probabilistic Programming
Probability Distribution
=
Logic Programs
+
Weights
[Fierens et al., TPLP’15]
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Probabilistic Programming
Probability Distribution
=
Logic Programs
+
Weights
+
path(X,Y) :-
edge(X,Y).
path(X,Y) :-
edge(X,Z), path(Z,Y).
[Fierens et al., TPLP’15]
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Conclusions
Relational probabilistic reasoning is frontier and integration of AI, KR, ML, DB, TH, etc.
We need – relational models and logic
– probabilistic models and statistical learning
– algorithms that scale
• Open-world data model – semantics make sense
– FREE for UCQs
– expensive otherwise
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Long-Term Outlook
Probabilistic inference and learning exploit
~ 1988: conditional independence
~ 2000: contextual independence (local structure)
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Long-Term Outlook
Probabilistic inference and learning exploit
~ 1988: conditional independence
~ 2000: contextual independence (local structure)
~ 201?: symmetry & exchangeability & first-order
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References
• Ceylan, Ismail Ilkan, Adnan Darwiche, and Guy Van den Broeck. "Open-world probabilistic databases." Proceedings of KR (2016).
• Suciu, Dan, Dan Olteanu, Christopher Ré, and Christoph Koch. "Probabilistic databases." Synthesis Lectures on Data Management 3, no. 2 (2011): 1-180.
• Dong, Xin, Evgeniy Gabrilovich, Geremy Heitz, Wilko Horn, Ni Lao, Kevin Murphy, Thomas Strohmann, Shaohua Sun, and Wei Zhang. "Knowledge vault: A web-scale approach to probabilistic knowledge fusion." In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 601-610. ACM, 2014.
• Carlson, Andrew, Justin Betteridge, Bryan Kisiel, Burr Settles, Estevam R. Hruschka Jr, and Tom M. Mitchell. "Toward an Architecture for Never-Ending Language Learning." In AAAI, vol. 5, p. 3. 2010.
• Niu, Feng, Ce Zhang, Christopher Ré, and Jude W. Shavlik. "DeepDive: Web-scale Knowledge-base Construction using Statistical Learning and Inference." VLDS 12 (2012): 25-28.
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References
• Chen, Brian X. "Siri, Alexa and Other Virtual Assistants Put to the Test" The New York Times (2016).
• Dalvi, Nilesh, and Dan Suciu. "The dichotomy of probabilistic inference for unions of conjunctive queries." Journal of the ACM (JACM) 59, no. 6 (2012): 30.
• De Raedt, Luc, Anton Dries, Ingo Thon, Guy Van den Broeck, and Mathias Verbeke. "Inducing probabilistic relational rules from probabilistic examples." In Proceedings of the 24th International Conference on Artificial Intelligence, pp. 1835-1843. AAAI Press, 2015.
• Van den Broeck, Guy. "Towards high-level probabilistic reasoning with lifted inference." AAAI Spring Symposium on KRR (2015).
• Niepert, Mathias, and Guy Van den Broeck. "Tractability through exchangeability: A new perspective on efficient probabilistic inference." AAAI (2014).
• Van den Broeck, Guy. "On the completeness of first-order knowledge compilation for lifted probabilistic inference." In Advances in Neural Information Processing Systems, pp. 1386-1394. 2011.
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References
• Van den Broeck, Guy, Wannes Meert, and Adnan Darwiche. "Skolemization for weighted first-order model counting." In Proceedings of the 14th International Conference on Principles of Knowledge Representation and Reasoning (KR). 2014.
• Gribkoff, Eric, Guy Van den Broeck, and Dan Suciu. "Understanding the complexity of lifted inference and asymmetric weighted model counting." UAI, 2014.
• Beame, Paul, Guy Van den Broeck, Eric Gribkoff, and Dan Suciu. "Symmetric weighted first-order model counting." In Proceedings of the 34th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pp. 313-328. ACM, 2015.
• Chavira, Mark, and Adnan Darwiche. "On probabilistic inference by weighted model counting." Artificial Intelligence 172.6 (2008): 772-799.
• Sang, Tian, Paul Beame, and Henry A. Kautz. "Performing Bayesian inference by weighted model counting." AAAI. Vol. 5. 2005.
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References
• Van den Broeck, Guy, Nima Taghipour, Wannes Meert, Jesse Davis, and Luc De Raedt. "Lifted probabilistic inference by first-order knowledge compilation." In Proceedings of the Twenty-Second international joint conference on Artificial Intelligence, pp. 2178-2185. AAAI Press/International Joint Conferences on Artificial Intelligence, 2011.
• Van den Broeck, Guy. Lifted inference and learning in statistical relational models. Diss. Ph. D. Dissertation, KU Leuven, 2013.
• Gogate, Vibhav, and Pedro Domingos. "Probabilistic theorem proving." UAI (2011).
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References
• Belle, Vaishak, Andrea Passerini, and Guy Van den Broeck. "Probabilistic inference in hybrid domains by weighted model integration." Proceedings of 24th International Joint Conference on Artificial Intelligence (IJCAI). 2015.
• Belle, Vaishak, Guy Van den Broeck, and Andrea Passerini. "Hashing-based approximate probabilistic inference in hybrid domains." In Proceedings of the 31st Conference on Uncertainty in Artificial Intelligence (UAI). 2015.
• Fierens, Daan, Guy Van den Broeck, Joris Renkens, Dimitar Shterionov, Bernd Gutmann, Ingo Thon, Gerda Janssens, and Luc De Raedt. "Inference and learning in probabilistic logic programs using weighted boolean formulas." Theory and Practice of Logic Programming 15, no. 03 (2015): 358-401.