Interative Hybrid Probabilistic Model Counting Steffen Michels, - PowerPoint PPT Presentation

Interative Hybrid Probabilistic Model Counting Steffen Michels, Arjen Hommersom, and Peter Lucas In proceedings of IJCAI 2016 Arjen Hommersom MultiLogic

Relational Probabilistic Problems Many probabilistic problems require hybrid reasoning have logical structure deal with rare observed events, e.g. diagnostic problems Representation of such problems: probabilistic logics capture and allow exploiting structure no direct support for hybrid reasoning can be extended with continuous distributions Arjen Hommersom MultiLogic

Probabilistic Logic Programming Knowledge base: Probabilistic Facts & Deterministic Rules (Sato’s Distribution Semantics) [Sato, 1995] Probabilistic Facts 0 . 2: low price ( apple ) Deterministic Rules (Closed-World Assumptions, ` a la Prolog ) buy ( Fruit ) ← low price ( Fruit ) means buy ( Fruit ) ⇐ ⇒ low price ( Fruit ) P ( buy ( apple )) = P ( low price ( Fruit )) = 0 . 2 Expressive enough for Bayesian Networks Exact inference feasible for many real worlds problems by transforming the problem into a weighted model counting (WMC) problem Arjen Hommersom MultiLogic

WMC: based on a DPLL-like procedure ( ϕ 1 ∨ ϕ 2 ) ∧ ( ϕ 1 ∨ ϕ 3 ) ¬ ϕ 1 ϕ 1 ( ϕ 1 ∨ ϕ 2 ) ∧ ( ϕ 1 ∨ ϕ 3 ) | ϕ 1 = true ( ϕ 1 ∨ ϕ 2 ) ∧ ( ϕ 1 ∨ ϕ 3 ) | ¬ ϕ 1 = ϕ 2 ∧ ϕ 3 ¬ ϕ 2 ϕ 2 ϕ 2 ∧ ϕ 3 | ϕ 2 = ϕ 3 ϕ 2 ∧ ϕ 3 | ¬ ϕ 2 = false ¬ ϕ 3 ϕ 3 ϕ 3 | ϕ 3 = true ϕ 3 | ¬ ϕ 3 = false Arjen Hommersom MultiLogic

WMC on this tree P ( ϕ 1 ) = 0 . 1 P ( ϕ 2 ) = 0 . 2 P ( ϕ 3 ) = 0 . 3 0 . 1 · 1 . 0 + 0 . 9 · 0 . 06 = 0 . 154 0 . 1 1 . 0 − 0 . 1 = 0 . 9 0 . 2 · 0 . 3 + 0 . 8 · 0 . 0 = 0 . 06 1 . 0 0 . 2 1 . 0 − 0 . 2 = 0 . 8 0 . 3 · 1 . 0 + 0 . 7 · 0 . 0 = 0 . 3 0 . 0 0 . 3 1 . 0 − 0 . 3 = 0 . 7 1 . 0 0 . 0 Arjen Hommersom MultiLogic

Hybrid Probabilistic Reasoning Hybrid probabilistic logic programs fails ( Comp ) ← FailCause ( Comp , Cause ) = true fails ( Comp ) ← Temp > Limit ( Comp ) fails ( Comp ) ← subcomp ( Subcomp , Comp ) , fails ( Subcomp ) FailCause ( engine , noFuel ) ∼ { 0 . 0002: true , 0 . 9998: false } Temp ∼ Γ(20 . 0 , 5 . 0) Limit ( engine ) ∼ N (65 . 0 , 5 . 0) subcomp ( fuelPump , engine ) Limit ( fuelPump ) ∼ N (80 . 0 , 5 . 0) · · · Probability of query event q , given evidence e : P ( q | e ) � � fails ( fuelPump ) | fails ( engine ) P How to do inference? Arjen Hommersom MultiLogic

Inference Methods Method Exact Rejection / MCMC IHPMC Importance Sampling Works finite prob- (virtually) (virtually) large class for lems only all problems all problems of hybrid problems Quality no error probabilistic none bounded er- guaran- ror tee Structure- yes no hand- yes sensitive tailored solution of- ten required Sensitive no yes no no to rare evidence Arjen Hommersom MultiLogic

IHPMC Basic Idea # � � P ( q ) = P ( q ) = P ( � ) # � + # � P ( q ) = P ( � ) + P ( � ) � P ( q ) = P ( � ) + P ( � ) / 2 ± P ( � ) / 2 Arjen Hommersom MultiLogic

Exploiting Structure Hybrid Probability Tree (HPT) A = true ∨ X > Y A = false A = true 0 . 9 0 . 1 P ( A = true ∨ X > Y ) ∈ [0 . 1 , 1 . 0] � ⊤ X > Y P ( A = true ∨ X > Y ) = 0 . 55 ± 0 . 45 Similar to binary WMC Exploits logical structure Search towards hyperrectangles with high probability Arjen Hommersom MultiLogic

Exploiting Structure Hybrid Probability Tree (HPT) A = true ∨ X > Y A = true A = false 0 . 9 0 . 1 P ( A = true ∨ X > Y ) ∈ [0 . 1 , 1 . 0] � ⊤ X > Y P ( A = true ∨ X > Y ) = 0 . 55 ± 0 . 45 X ∈ ( −∞ , 20] X ∈ (20 , ∞ ) 0 . 8 0 . 2 P ( A = true ∨ X > Y ) ∈ [0 . 1 , 1 . 0] � X > Y X > Y P ( A = true ∨ X > Y ) = 0 . 55 ± 0 . 45 Similar to binary WMC Exploits logical structure Search towards hyperrectangles with high probability Arjen Hommersom MultiLogic

Exploiting Structure Hybrid Probability Tree (HPT) A = true ∨ X > Y A = false A = true 0 . 1 0 . 9 P ( A = true ∨ X > Y ) ∈ [0 . 1 , 1 . 0] � ⊤ X > Y P ( A = true ∨ X > Y ) = 0 . 55 ± 0 . 45 X ∈ ( −∞ , 20] X ∈ (20 , ∞ ) 0 . 8 0 . 2 P ( A = true ∨ X > Y ) ∈ [0 . 1 , 1 . 0] � X > Y X > Y P ( A = true ∨ X > Y ) = 0 . 55 ± 0 . 45 Y ∈ ( −∞ , 20] Y ∈ (20 , ∞ ) 0 . 1 0 . 9 P ( A = true ∨ X > Y ) ∈ [0 . 1 , 0 . 352] � X > Y ⊥ P ( A = true ∨ X > Y ) = 0 . 226 ± 0 . 126 Similar to binary WMC Exploits logical structure Search towards hyperrectangles with high probability Arjen Hommersom MultiLogic

Theoretical property Approximations with arbitrary precision can be computed For all events q and e and every maximal error ǫ , IPHMC can in finite time find an approximation such that: P ( q | e ) − P ( q | e ) ≤ ǫ and P ( q | e ) − P ( q | e ) ≤ ǫ Arjen Hommersom MultiLogic

No Evidence 9.0e -4 IHPMC BLOG LW BLOG RJ DC (Mean) Squarred Error 6.0e -4 3.0e -4 0.0e -4 0.0 0.2 0.4 0.6 0.8 1.0 Inference Time (s) P ( fails (9)), p = 0 . 01, µ = 60 . 0 Arjen Hommersom MultiLogic

Rare Observed Event 6.0e -2 IHPMC BLOG LW BLOG RJ DC (Mean) Squarred Error 4.0e -2 2.0e -2 0.0e -2 0.0 0.5 1.0 1.5 2.0 Inference Time (s) P ( fails (9) | fails (0)), p = 0 . 0001, µ = 60 . 0 Arjen Hommersom MultiLogic

Conclusions IHPMC provides alternative to sampling insensitive to rare observed events no hand-tailoring bounded error may fail, but lets the user know! Try it: http://www.steffen-michels.de/ihpmc Arjen Hommersom MultiLogic