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

ISIPTA 2019 GHENT 03/07 ALESSANDRO ANTONUCCI, ALESSANDRO FACCHINI & LILITH MATTEI CREDAL SENTENTIAL DECISION DIAGRAMS

▸ Alessandro Antonucci, a senior researcher in probabilistic graphical models and machine learning Istituto Dalle Molle di Studi per l’Intelligenza Artificiale ▸ Alessandro Facchini, a ▸ Lilith Mattei, research assistant, convenience* logician wannabe PhD student *Concept and formulation by Yoichi Hirai

FRAMING THE PROBLEM - STATE OF THE ART WHAT ARE CSDD? FIRST SOME ZOOLOGY

FRAMING THE PROBLEM - STATE OF THE ART WHAT ARE CSDD? FIRST SOME ZOOLOGY Bayesian nets (Pearl, 1984)

FRAMING THE PROBLEM - STATE OF THE ART WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984)

FRAMING THE PROBLEM - STATE OF THE ART WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984) Tractable “deep” model ? Sum-product nets (Poon & Domingos, 2012)

FRAMING THE PROBLEM - STATE OF THE ART WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984) Tractable “deep” model ? Sum-product nets Imprecise version? Credal SPNs (Poon & Domingos, (Mauá et al., 2012) 2017)

FRAMING THE PROBLEM - STATE OF THE ART WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984) Tractable “deep” model ? Sum-product nets Imprecise version? Credal SPNs (Poon & Domingos, (Mauá et al., 2012) 2017) …and logical constraints ? Probabilistic Sentential Decision Diagrams, PSDDs (Kisa et al., 2014)

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

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

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

FRAMING THE PROBLEM WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984) Tractable “deep” model ? Sum-product nets Imprecise version? Credal SPNs (Poon & Domingos, (Mauá et al., 2012) 2017) Do nice properties of CSPNs adapt to CSDD? …and logical constraints ? YES ! Imprecise version? Probabilistic Sentential CSDD (here) Decision Diagrams, PSDDs (Kisa et al., 2014) ‣ Fast marginal inference algorithm for general CSPNs ‣ Fast conditional inference algorithm for singly connected CSPNs

FRAMING THE PROBLEM WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984) Tractable “deep” model ? Sum-product nets Imprecise version? Credal SPNs (Poon & Domingos, (Mauá et al., 2012) 2017) Do nice properties of CSPNs adapt to CSDD? …and logical constraints ? YES ! Imprecise version? Probabilistic Sentential CSDD (here) Decision Diagrams, PSDDs (Kisa et al., 2014) message of this work: CSDD’s stand to PSDD’s as CSPN's stand to SPN’s

FRAMING THE PROBLEM WHAT ARE CSDD? FIRST SOME ZOOLOGY Imprecise version? Credal nets Bayesian nets (Cozman, 2000) (Pearl, 1984) Tractable “deep” model ? Sum-product nets Imprecise version? Credal SPNs (Poon & Domingos, (Mauá et al., 2012) 2017) Do nice properties of CSPNs adapt to CSDD? …and logical constraints ? YES ! Imprecise version? Probabilistic Sentential CSDDs (here) Decision Diagrams, PSDDs (Kisa et al., 2014) message of this work: CSDD’s stand to PSDD’s as …but what are CSDDs ? CSPN's stand to SPN’s

WHAT ARE CSDD’S ? A FIRST GLIMPSE TO CSDD

WHAT ARE CSDD’S ? A FIRST GLIMPSE TO CSDD ‣ CSDD = Credal version of Probabilistic Sentential Decision Diagrams

WHAT ARE CSDD’S ? A FIRST GLIMPSE TO CSDD ‣ CSDD = Credal version of Probabilistic Sentential Decision Diagrams ▸ so, what are PSDDs?

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?

TOY EXAMPLE (FROM KISA ET AL. 2014) 100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A) L K P A ▸ 16 joint states 0 0 0 0 0 0 0 1 0 0 1 0 ▸ Three logical constraints 0 0 1 1 0 1 0 0 0 1 0 1 ( P ∨ L ), ( A → P ), ( K → A ∨ L ) 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1

TOY EXAMPLE (FROM KISA ET AL. 2014) 100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A) L K P A ▸ 16 joint states 0 0 0 0 0 0 0 1 0 0 1 0 ▸ Three logical constraints 0 0 1 1 0 1 0 0 0 1 0 1 ϕ := ( P ∨ L ) ∧ ( A → P ) ∧ ( K → A ∨ L ) 0 1 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1

TOY EXAMPLE (FROM KISA ET AL. 2014) 100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A) K P A L ▸ 16 joint states 0 0 0 0 0 0 0 1 0 0 1 0 ▸ Three logical constraints 0 0 1 1 0 1 0 0 ϕ := ( P ∨ L ) ∧ ( A → P ) ∧ ( K → A ∨ L ) 0 1 0 1 0 1 1 0 ▸ 7 states not satisfying the 0 1 1 1 1 0 0 0 logical constraints (hence 1 0 0 1 never observed) 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1

TOY EXAMPLE (FROM KISA ET AL. 2014) 100 STUDENTS ENROLLING IN 4 CLASSES: LOGIC (L), KNOWLEDGE REPRESENTATION (K), PROBABILITY (P), AI (A) K P A L ▸ 16 joint states 0 0 0 0 0 0 0 1 0 0 1 0 ▸ Three logical constraints 0 0 1 1 0 1 0 0 ϕ := ( P ∨ L ) ∧ ( A → P ) ∧ ( K → A ∨ L ) 0 1 0 1 0 1 1 0 ▸ 7 states not satisfying the 0 1 1 1 logical constraints (hence 1 0 0 0 never observed) 1 0 0 1 1 0 1 0 1 0 1 1 ▸ 1 state logically possible but 1 1 0 0 never observed 1 1 0 1 1 1 1 0 1 1 1 1

CONSTRAINTS FIRST: SDDS MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011) A Sentential Decision Diagram representing is a “deterministic” logic circuit ▸ ϕ

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 )

CONSTRAINTS FIRST: SDDS MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011) A Sentential Decision Diagram representing is a “deterministic” logic circuit ▸ ϕ ▸ Partition ⊤ = (¬ L ∧ K ) ∨ L ∨ (¬ L ∧ ¬ K )

CONSTRAINTS FIRST: SDDS MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011) A Sentential Decision Diagram representing is a “deterministic” logic circuit ▸ ϕ take a subset of the variables, form a partition of the tautology, e.g., ▸ ⊤ = (¬ L ∧ K ) ∨ L ∨ (¬ L ∧ ¬ K ) L ¬ L ∧ ¬ K ¬ L ∧ K ϕ = ( P ∨ L ) ∧ ( P ∨ ¬ A ) ∧ ( A ∨ L ∨ ¬ K )

CONSTRAINTS FIRST: SDDS MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011) A Sentential Decision Diagram representing is a “deterministic” logic circuit ▸ ϕ take a subset of the variables, form a partition of the tautology, e.g., ▸ ⊤ = (¬ L ∧ K ) ∨ L ∨ (¬ L ∧ ¬ K ) L ¬ L ∧ ¬ K ¬ L ∧ K ϕ What does becomes when L = ⊥ , K = ⊤ ? ϕ = ( P ∨ L ) ∧ ( P ∨ ¬ A ) ∧ ( A ∨ L ∨ ¬ K )

CONSTRAINTS FIRST: SDDS MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011) A Sentential Decision Diagram representing is a “deterministic” logic circuit ▸ ϕ take a subset of the variables, form a partition of the tautology, e.g., ▸ ⊤ = (¬ L ∧ K ) ∨ L ∨ (¬ L ∧ ¬ K ) L ¬ L ∧ ¬ K ¬ L ∧ K P ∧ A ϕ = ( P ∨ L ) ∧ ( P ∨ ¬ A ) ∧ ( A ∨ L ∨ ¬ K )

CONSTRAINTS FIRST: SDDS MODELING CONSTRAINTS WITH CIRCUITS: SDD’S (DARWICHE 2011) A Sentential Decision Diagram representing is a “deterministic” logic circuit ▸ ϕ take a subset of the variables, form a partition of the tautology, e.g., ▸ ⊤ = (¬ L ∧ K ) ∨ L ∨ (¬ L ∧ ¬ K ) L ¬ L ∧ ¬ K ¬ L ∧ K ϕ ϕ What does What does P ∧ A becomes when becomes when L = ⊤ ? L = ⊥ , K = ⊥ ? ϕ = ( P ∨ L ) ∧ ( P ∨ ¬ A ) ∧ ( A ∨ L ∨ ¬ K )