[PPT] - CREDAL SENTENTIAL DECISION DIAGRAMS Alessandro Antonucci, a PowerPoint Presentation

SLIDE 1

CREDAL SENTENTIAL DECISION DIAGRAMS

ALESSANDRO ANTONUCCI, ALESSANDRO FACCHINI & LILITH MATTEI

ISIPTA 2019 GHENT 03/07

SLIDE 2

Istituto Dalle Molle di Studi per l’Intelligenza Artificiale

▸ Alessandro

Antonucci, a senior researcher in probabilistic graphical models and machine learning

▸ Alessandro Facchini, a

convenience* logician

*Concept and formulation by Yoichi Hirai

▸ Lilith Mattei, research assistant,

wannabe PhD student

SLIDE 3

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

SLIDE 4

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984)

SLIDE 5

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Imprecise version?

SLIDE 6

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Imprecise version? Sum-product nets (Poon & Domingos, 2012)

SLIDE 7

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) Imprecise version? Imprecise version? Sum-product nets (Poon & Domingos, 2012)

SLIDE 8

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Sum-product nets (Poon & Domingos, 2012) Credal SPNs (Mauá et al., 2017) …and logical constraints ? Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014) Imprecise version? Imprecise version?

SLIDE 9

FRAMING THE PROBLEM - STATE OF THE ART

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) …and logical constraints ? CSDD (here) Imprecise version? Imprecise version? Imprecise version? Sum-product nets (Poon & Domingos, 2012) Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014)

SLIDE 10

FRAMING THE PROBLEM

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) …and logical constraints ? CSDD (here) Imprecise version? Imprecise version? Imprecise version? ? Do nice properties of CSPNs adapt to CSDD? Sum-product nets (Poon & Domingos, 2012) Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014)

SLIDE 11

FRAMING THE PROBLEM

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) …and logical constraints ? CSDD (here) Imprecise version? Imprecise version? Imprecise version? Do nice properties of CSPNs adapt to CSDD? YES ! Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014) Sum-product nets (Poon & Domingos, 2012)

SLIDE 12

FRAMING THE PROBLEM

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) …and logical constraints ? CSDD (here) Imprecise version? Imprecise version? Imprecise version?

Fast marginal inference algorithm for

general CSPNs

Fast conditional inference algorithm

for singly connected CSPNs

Do nice properties of CSPNs adapt to CSDD? YES ! Sum-product nets (Poon & Domingos, 2012) Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014)

SLIDE 13

FRAMING THE PROBLEM

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) …and logical constraints ? CSDD (here) Imprecise version? Imprecise version? Imprecise version?

message of this work: CSDD’s stand to PSDD’s as CSPN's stand to SPN’s

Do nice properties of CSPNs adapt to CSDD? YES ! Sum-product nets (Poon & Domingos, 2012) Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014)

SLIDE 14

FRAMING THE PROBLEM

WHAT ARE CSDD? FIRST SOME ZOOLOGY

Bayesian nets (Pearl, 1984) Credal nets (Cozman, 2000) Tractable “deep” model ? Credal SPNs (Mauá et al., 2017) …and logical constraints ? CSDDs (here) Imprecise version? Imprecise version? Imprecise version?

…but what are CSDDs ?

message of this work: CSDD’s stand to PSDD’s as CSPN's stand to SPN’s

Do nice properties of CSPNs adapt to CSDD? YES ! Sum-product nets (Poon & Domingos, 2012) Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014)

SLIDE 15

WHAT ARE CSDD’S ?

A FIRST GLIMPSE TO CSDD

SLIDE 16

WHAT ARE CSDD’S ?

A FIRST GLIMPSE TO CSDD

CSDD = Credal version of Probabilistic Sentential

Decision Diagrams

SLIDE 17

WHAT ARE CSDD’S ?

A FIRST GLIMPSE TO CSDD

CSDD = Credal version of Probabilistic Sentential

Decision Diagrams

▸ so, what are PSDDs?

SLIDE 18

WHAT ARE CSDD’S ?

A FIRST GLIMPSE TO CSDD

CSDD = Credal version of Probabilistic Sentential

Decision Diagrams

▸ so, what are PSDDs? ▸ actually, what are SDDs?

SLIDE 19

TOY EXAMPLE (FROM KISA ET AL. 2014)

100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A)

▸ 16 joint states ▸ Three logical constraints

L K P A

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

(P ∨ L), (A → P), (K → A ∨ L)

SLIDE 20

TOY EXAMPLE (FROM KISA ET AL. 2014)

▸ 16 joint states ▸ Three logical constraints

L K P A

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

ϕ := (P ∨ L) ∧ (A → P) ∧ (K → A ∨ L)

100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A)

SLIDE 21

TOY EXAMPLE (FROM KISA ET AL. 2014)

▸ 16 joint states ▸ Three logical constraints ▸ 7 states not satisfying the

logical constraints (hence never observed)

L

K P A

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

ϕ := (P ∨ L) ∧ (A → P) ∧ (K → A ∨ L)

100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A)

SLIDE 22

TOY EXAMPLE (FROM KISA ET AL. 2014)

▸ 16 joint states ▸ Three logical constraints ▸ 7 states not satisfying the

logical constraints (hence never observed)

▸ 1 state logically possible but

never observed

L

K P A

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

ϕ := (P ∨ L) ∧ (A → P) ∧ (K → A ∨ L)

100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A)

SLIDE 23

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

ϕ

SLIDE 24

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

ϕ

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

SLIDE 25

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

ϕ

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

Partition

SLIDE 26

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

ϕ

¬L ∧ K L ¬L ∧ ¬K

ϕ = (P ∨ L) ∧ (P ∨ ¬A) ∧ (A ∨ L ∨ ¬K)

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

SLIDE 27

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

ϕ

¬L ∧ K L ¬L ∧ ¬K

ϕ = (P ∨ L) ∧ (P ∨ ¬A) ∧ (A ∨ L ∨ ¬K)

What does becomes when

ϕ

L = ⊥ , K = ⊤ ?

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

SLIDE 28

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

ϕ

¬L ∧ K L ¬L ∧ ¬K P ∧ A

ϕ = (P ∨ L) ∧ (P ∨ ¬A) ∧ (A ∨ L ∨ ¬K)

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

SLIDE 29

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

ϕ

¬L ∧ K L ¬L ∧ ¬K

ϕ = (P ∨ L) ∧ (P ∨ ¬A) ∧ (A ∨ L ∨ ¬K)

What does becomes when

ϕ

L = ⊤ ? What does becomes when

ϕ

L = ⊥ , K = ⊥ ?

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

P ∧ A

SLIDE 30

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

ϕ

¬L ∧ K L ¬L ∧ ¬K P ∧ A

ϕ = (P ∨ L) ∧ (P ∨ ¬A) ∧ (A ∨ L ∨ ¬K)

P ∨ ¬A

P

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

SLIDE 31

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

▸

ϕ

¬L ∧ K L ¬L ∧ ¬K P ∧ A P ∨ ¬A

P

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

(¬L ∧ K) ∧ (P ∧ A)⋁L ∧ (P ∨ ¬A)⋁(¬L ∧ ¬K) ∧ P = ϕ

SLIDE 32

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

▸

A Sentential Decision Diagram representing is a “deterministic” logic circuit

▸

take a subset of the variables, form a partition of the tautology, e.g.,

▸ ▸

Proceed recursively…

ϕ

¬L ∧ K L ¬L ∧ ¬K P ∧ A P ∨ ¬A

P

⊤ = (¬L ∧ K) ∨ L ∨ (¬L ∧ ¬K)

(¬L ∧ K) ∧ (P ∧ A)⋁L ∧ (P ∨ ¬A)⋁(¬L ∧ ¬K) ∧ P = ϕ

SLIDE 33

CONSTRAINTS FIRST: SDDS

MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011)

(¬L ∧ K⋁L ∧ ⊥ )⋀(P ∧ A⋁¬P ∧ ⊥ )

(L ∧ ⊤ ⋁¬L ∧ ⊥ )⋀(¬P ∧ ¬A⋁P ∧ ⊤ )

(¬L ∧ ¬K⋁L ∧ ⊥ )⋀(P ∧ ⊤ ⋁¬P ∧ ⊥ )

∨ ∨

= ϕ

Paired boxes: AND gates Decision nodes: OR gates

SLIDE 34

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

ϕ

SLIDE 35

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

▸ Parameters learned from data

ϕ

SLIDE 36

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

▸ Parameters learned from data ▸ Inducing a joint probability

ℙ(A, L, P, K)

ϕ

SLIDE 37

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

▸ Parameters learned from data ▸ Inducing a joint probability

ℙ(A, L, P, K)

ϕ

ℙ(L ∧ ¬K)

SLIDE 38

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

▸ Parameters learned from data ▸ Inducing a joint probability

ℙ(A, L, P, K)

ϕ

ℙ(P|L)

ℙ(L ∧ ¬K)

SLIDE 39

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

▸ Parameters learned from data ▸ Inducing a joint probability ▸ context-specific independences wrt derived from the structure

ℙ(A, L, P, K)

ℙ ϕ

ℙ(P|L)

ℙ(L ∧ ¬K)

SLIDE 40

CONSTRAINTS FIRST, DATA AFTER: PSDD

MODELING DATA + CONSTRAINTS WITH CIRCUITS: PSDD’S (KISA, 2014)

▸ A Probabilistic Sentential Decision Diagrams (PSDDs) for. is a

parametrized SDD:

▸ Parameters learned from data ▸ Inducing a joint probability ▸ context-specific independences wrt derived from the structure ▸ Logically impossible events have zero probability:

ℙ(A, L, P, K)

ℙ(x) > 0 ↔ x ⊧ ϕ

ℙ ϕ

ℙ(P|L)

ℙ(L ∧ ¬K)

SLIDE 41

DEFINING CSDD’S

CREDAL VERSION OF PSDD’S:

SLIDE 42

DEFINING CSDD’S

CREDAL VERSION OF PSDD’S: REPLACE PMF’S WITH CS’S

SLIDE 43

DEFINING CSDD’S

CREDAL VERSION OF PSDD’S: REPLACE PMF’S WITH CS’S

ϕ

▸ Credal Sentential Decision Diagrams (CSDDs) for

[ 10

101 , 11 101 ]

[ 30

101 , 31 101 ]

[ 60

101 , 61 101 ]

[ 24

31 , 25 31 ]

[ 3

13 , 4 13 ]

[ 54

61 , 55 61 ]

[ 18

31 , 19 31 ]

[ 12

31 , 13 31 ]

SLIDE 44

DEFINING CSDD’S

CREDAL VERSION OF PSDD’S: REPLACE PMF’S WITH CS’S

ϕ

▸ Credal Sentential Decision Diagrams (CSDDs) for ▸ Syntax: CS attached to each decision node and to each terminal node

⊤

[ 10

101 , 11 101 ]

[ 30

101 , 31 101 ]

[ 60

101 , 61 101 ]

[ 24

31 , 25 31 ]

[ 3

13 , 4 13 ]

[ 54

61 , 55 61 ]

[ 18

31 , 19 31 ]

[ 12

31 , 13 31 ]

SLIDE 45

DEFINING CSDD’S

CREDAL VERSION OF PSDD’S: REPLACE PMF’S WITH CS’S

ϕ

▸ Credal Sentential Decision Diagrams (CSDDs) for ▸ Syntax: CS attached to each decision node and to each terminal node ▸ Semantics: collection of consistent PSDDs

⊤

[ 10

101 , 11 101 ]

[ 30

101 , 31 101 ]

[ 60

101 , 61 101 ]

[ 24

31 , 25 31 ]

[ 3

13 , 4 13 ]

[ 54

61 , 55 61 ]

[ 18

31 , 19 31 ]

[ 12

31 , 13 31 ]

SLIDE 46

DEFINING CSDD’S

CREDAL VERSION OF PSDD’S: REPLACE PMF’S WITH CS’S

ϕ

▸ Credal Sentential Decision Diagrams (CSDDs) for ▸ Syntax: CS attached to each decision node and to each terminal node ▸ Semantics: collection of consistent PSDDs ▸ PSDD induces joint P, CSDD induces joint CS (“Strong extension”)

⊤

[ 10

101 , 11 101 ]

[ 30

101 , 31 101 ]

[ 60

101 , 61 101 ]

[ 24

31 , 25 31 ]

[ 3

13 , 4 13 ]

[ 54

61 , 55 61 ]

[ 18

31 , 19 31 ]

[ 12

31 , 13 31 ]

SLIDE 47

CSDD’S INFERENCE

SLIDE 48

CSDD’S INFERENCE

Marginal queries:

Given evidence e, calculate

ℙ(e) = min

ℙ(X)∈𝕃(X) ℙ(e)

ℙ(x|e) = min

ℙ(X)∈𝕃(X)

ℙ(x, e) ℙ(e)

SLIDE 49

CSDD’S INFERENCE

Marginal queries:

Given evidence e, calculate

ℙ(e) = min

ℙ(X)∈𝕃(X) ℙ(e)

Conditional queries:

Given available evidence e and queried variabile, calculate

ℙ(x|e) = min

ℙ(X)∈𝕃(X)

ℙ(x, e) ℙ(e)

SLIDE 50

TWO POLYTIME ALGORITHMS

▸ Adaptation of CSPNs algorithms (Mauá et al.) to CSDDs:

Marginal queries:

Bottom-up propagation of LP

task’s results

Coefficients of each LP task are

computed in the lower level

Feasible regions are the local

CSs

Conditional queries:

Decisional version of original task
Bottom-up propagation of LP task’s

results

Coefficients of each LP task are

computed in the lower level, depending on evidence

Feasible regions are the local CSs

CSDD’S INFERENCE

SLIDE 51

TWO POLYTIME ALGORITHMS

▸ Adaptation of CSPNs algorithms (Mauá et al.) to CSDDs:

Marginal queries:

Bottom-up propagation of LP

task’s results

Coefficients of each LP task are

computed in the lower level

Feasible regions are the local

CSs

Conditional queries:

Decisional version of original task
Bottom-up propagation of LP task’s

results

Coefficients of each LP task are

computed in the lower level, depending on evidence

Feasible regions are the local CSs

CSDD’S INFERENCE

Needs singly connected topology

SLIDE 52

CONCLUSIONS AND FUTURE WORK

▸ CSDDs as a new tool for sensitivity analysis in PSDD ▸ Robust marginalisation and conditioning (for singly connected

circuits) with poly complexity ▸ Application to “credal” ML with structured spaces ▸ Complexity and approximations results for multiply connected

CSDDs

▸ Hybrid (structured/unstructured) models ▸ Structural learning (trade-off small SDD / likelihood / independences) ▸ CNs vs. CSDDs ?

SLIDE 53

CONCLUSIONS AND FUTURE WORK

▸ CSDDs as a new tool for sensitivity analysis in PSDD ▸ Robust marginalisation and conditioning (for singly connected

circuits) with poly complexity ▸ Application to “credal” ML with structured spaces ▸ Complexity and approximations results for multiply connected

CSDDs

▸ Hybrid (structured/unstructured) models ▸ Structural learning (trade-off small SDD / likelihood / independences) ▸ CNs vs. CSDDs ?

SLIDE 54

Credal Sentential Decision Diagrams (CSDDs)

Alessandro Antonucci, Alessandro Facchini, Lilith Mattei

{alessandro, alessandro.facchini, lilith}@idsia.ch A NEW CLASS OF (CREDAL) GRAPHICAL MODELS

Bayesian nets as classical (precise) probabilistic graphical models (BNs)
With imprecise probabilities? Credal networks (CNs, Cozman, 2000)
With deep structure (and tractable inference)?

Sum-product networks (SPNs, Poon & Domingos, 2011)

With deep structure and imprecise probabilities?

Credal sum-product networks (CSPNs, Mauá et al., 2017)

With deep structure and embedding logical constraints?

Probabilistic sentential decision diagrams (PSDDs, Kisa et al., 2014)

Deep structure, imprecise probabilities and logical constraints?

Credal sentential decision diagrams (CSDDs, this paper) TOY EXAMPLE: CLASSES ENROLLMENT

Data about 100 students in four classes
Logic, Knowledge, Probability and

Artificial Intelligence

Logical constraints for classes:

φ := (P _ L) ^ (A ! P) ^ (K ! A _ L)

Out of 24 = 16 joint configurations,
nly eight in the data set

seven are logically impossible,

ne possible but observed)
Robust learning of a model over (L,K,P,A)?
Consistent with the logical constraints φ?
The solution is a CSSD!

L K P A # 1 1 6 1 1 54 1 1 1 1 1 1 1 1 10 1 5 1 1 1 1 1 1 1 1 1 1 13 1 1 1 1 1 1 8 1 1 1 1 3 FROM SDDS TO CSDDS (THROUGH PSDDS)

Logical skeleton? φ as a circuit alternating OR and AND gates
This is a sentential decision diagram, (SDD, Choi & Darwiche, 2013)
Probabilistic model? Probability mass functions annotating

the OR gates of the SDD (PSDDs)

PSDD is a joint probability mass function consistent with the constraints

P(L, K, P, A) : P(l, k, p, a) = 0 iff (l, k, p, a) 6| = φ

CSDD? Credal version of PSDD: credal sets instead of mass functions
Credal sets on OR gates and terminal nodes >
Semantics: all PSDDs with parameters consistent with the local credal sets
Strong extension K(L, K, P, A) as the joint credal set of

all the joint mass functions induced by the consistent PSDDs

CSDD Inference? Lower/upper bounds wrt the strong extension
Base theorem: for each z: P(z) > 0 iff z |

= φ and P(z) = 0 iff z 6| = φ

Learning CSDD? Parameters are conditional probabilities,

Imprecise Dirichlet Model to learn local (conditional) credal sets

Data scarcity issue on the leaves.justifies imprecise approach!

MARGINAL QUERIES

Circuit traversal from leaves in re-

verse topological order

Every time a decision node is pro-

cessed, a LP task whose feasible region are the local credal sets of the node should be solved.

Analogous to Mauá et al. (2017)

for CSPNs, with additionally sup- port to logical constraints CONDITIONAL QUERIES

Conditional queries solved by generalized Bayes’ rule (GBR)
Associated decision problem is deciding whether or not,

for a given µ 2 [0, 1]: P(x|e) > µ

As P(x|e) + P(¬x|e) = 1 for each P(X) 2 Kr(X),

and assuming that P(e) > 0, this corresponds to: minP(X)2Kr(X) [(1 µ)P(x, e) µP(¬x, e)] > 0

Recursive formulation (for singly connected circuits):

min[θ1,...,θk]2Kr(P) ∑k i=1 π(pi) σ(si) θi > 0

where π(pi) is equal to minPpi (Z)2Kpi (Z)

⇥(1 µ)Ppi(x, el) µPpi(¬x, el) ⇤

and σ(si) is equal to

( Psi(er) if π(pi) < 0 Psi(er)

therwise.
Circuit

traversal from leaves (as for marginal queries)

LP tasks on deci-

sion nodes whose coefficients are computed with marginal queries

Bracketing

scheme to solve GBR

Again analogous

to Mauá et al. (2017) result for CSPNs CONCLUSIONS & OUTLOOKS

CSDDs as a new tool for sensitivity analysis in PSDD
Fast robust marginalisation and conditioning

(but conditioning works for singly connected circuits only)

Complexity results and approximated algorithm are needed
CNs vs. CSDDs? Credal classification with CSDDs?

REFERENCES

Hoifung Poon & Pedro Domingos. Sum-product networks: a new deep architecture.

In IEEE ICCV Workshops, pages 689-690. IEEE, 2011.

Denis Mauá, Fabio Cozman, Diarmaid Conaty, and Cassio de Campos. Credal sum-

product networks. In Proceedings of ISIPTA ’17, pages 205-216, 2017.

Denis Deratani Mauá, Diarmaid Conaty, Fabio Cozman, Katja Poppenhaeger, and

Cassio de Campos. Robustifying sum-product networks. International Journal of Approximate Reasoning, 2018.

Doga Kisa, Guy Van den Broeck, Arthur Choi, and Adnan Darwiche. Probabilistic

sentential decision diagrams. In Proceedings of the Fourteenth International Confer- ence on the Principles of Knowledge Representation and Reasoning, 2014.

Fabio Cozman. Credal networks. Artificial Intelligence, 120:199-233, 2000.
Arthur Choi and Adnan Darwiche. Dynamic minimization of sentential decision
diagrams. In Proceedings of the Twenty-Seventh AAAI Conference on Artificial In-

telligence, 2013.