Valuation Algebras Local Computation Semiring Induced Valuation Algebras Optimization as Inference Problem An Introduction to Valuation Algebras Marc Pouly [email protected] Department of Informatics University of Fribourg, Switzerland Leukerbad Meeting January 2007 Marc Pouly An Introduction to Valuation Algebras 1/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Outline 1 Valuation Algebras Unifying Knowledge Representation Systems Valuation Algebras 2 Local Computation 3 Semiring Induced Valuation Algebras 4 Optimization as Inference Problem Marc Pouly An Introduction to Valuation Algebras 2/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Knowledge Representation Systems There are many ways to represent knowledge and information. Bayesian Possibility Belief Credal Networks Potentials Functions Sets Indicator Boolean Constraint Relations Functions Functions Systems Equality Gaussian Logics Systems Potentials ... Traditionally, every research field has its own community. People rarely look beyond their own nose ... Marc Pouly An Introduction to Valuation Algebras 3/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Knowledge Representation Systems There are many ways to represent knowledge and information. Bayesian Possibility Belief Credal Networks Potentials Functions Sets Indicator Boolean Constraint Relations Functions Functions Systems Equality Gaussian Logics Systems Potentials ... Traditionally, every research field has its own community. People rarely look beyond their own nose ... Marc Pouly An Introduction to Valuation Algebras 3/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Requirements for a unifying Theory Is there a unifying theory for all these formalisms ? Informal requirements to such a theory: Knowledge exists in pieces ! valuations. Knowledge refers to questions ! domain. Pieces of knowledge can be combined. Knowledge can be focused (marginalized). Marc Pouly An Introduction to Valuation Algebras 4/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Requirements for a unifying Theory Is there a unifying theory for all these formalisms ? Informal requirements to such a theory: Knowledge exists in pieces ! valuations. Knowledge refers to questions ! domain. Pieces of knowledge can be combined. Knowledge can be focused (marginalized). Marc Pouly An Introduction to Valuation Algebras 4/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Operations on Valuations Knowledge always refers to a domain: 1 Labeling: Φ D; φ d(φ) → "→ Pieces of knowledge can be combined: 2 Combination: Φ Φ Φ; (φ, ψ) φ ψ × → "→ ⊗ Knowledge can be focused: 3 Marginalization: Φ D Φ; (φ, x) φ x × → "→ ↓ Marginalization ”can” be replaced by: X d(φ) X 3 Variable Elimination: φ− = φ↓ −{ }. Marc Pouly An Introduction to Valuation Algebras 5/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Operations on Valuations Knowledge always refers to a domain: 1 Labeling: Φ D; φ d(φ) → "→ Pieces of knowledge can be combined: 2 Combination: Φ Φ Φ; (φ, ψ) φ ψ × → "→ ⊗ Knowledge can be focused: 3 Marginalization: Φ D Φ; (φ, x) φ x × → "→ ↓ Marginalization ”can” be replaced by: X d(φ) X 3 Variable Elimination: φ− = φ↓ −{ }. Marc Pouly An Introduction to Valuation Algebras 5/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Case Study I: Probability Potentials Burglary A B B 0 0 0.8 φ = 0 0.9 ψ = 0 1 0.1 1 0.1 1 0 0.2 Alarm 1 1 0.9 Combination = Multiplication Marginalization = Addition A B 0 0 0.72 A φ ψ = 0 1 0.01 (φ ψ) A = 0 0.73 ⊗ ⊗ ↓{ } 1 0 0.18 1 0.27 1 1 0.09 Marc Pouly An Introduction to Valuation Algebras 6/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Case Study I: Probability Potentials Burglary A B B 0 0 0.8 φ = 0 0.9 ψ = 0 1 0.1 1 0.1 1 0 0.2 Alarm 1 1 0.9 Combination = Multiplication Marginalization = Addition A B 0 0 0.72 A φ ψ = 0 1 0.01 (φ ψ) A = 0 0.73 ⊗ ⊗ ↓{ } 1 0 0.18 1 0.27 1 1 0.09 Marc Pouly An Introduction to Valuation Algebras 6/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Case Study II: Relations Player Club Goals Player Nationality Ronaldinho Barcelona 7 Ronaldinho Brazil φ = Eto’o Barcelona 5 ψ = Eto’o Cameroon Henry Arsenal 5 Henry France Pires Arsenal 2 Pires France Combination = Natural Join Player Club Goals Nationality Ronaldinho Barcelona 7 Brazil φ ψ = Eto’o Barcelona 5 Cameroon ⊗ Henry Arsenal 5 France Pires Arsenal 2 France Marginalization = Projection Goals Nationality Goals, Nationality 7 Brazil (φ ψ)↓{ } = 5 Cameroon ⊗ 5 France 2 France Marc Pouly An Introduction to Valuation Algebras 7/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Case Study II: Relations Player Club Goals Player Nationality Ronaldinho Barcelona 7 Ronaldinho Brazil φ = Eto’o Barcelona 5 ψ = Eto’o Cameroon Henry Arsenal 5 Henry France Pires Arsenal 2 Pires France Combination = Natural Join Player Club Goals Nationality Ronaldinho Barcelona 7 Brazil φ ψ = Eto’o Barcelona 5 Cameroon ⊗ Henry Arsenal 5 France Pires Arsenal 2 France Marginalization = Projection Goals Nationality Goals, Nationality 7 Brazil (φ ψ)↓{ } = 5 Cameroon ⊗ 5 France 2 France Marc Pouly An Introduction to Valuation Algebras 7/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Axioms of a Valuation Algebra 1 Commutative Semigroup: is associative & commutative. ⊗ 2 Labeling: d(φ ψ) = d(φ) d(ψ). ⊗ ∪ 3 Marginalization: x d(φ↓ ) = x. 4 Transitivity: y x x For x y d(φ): (φ↓ )↓ = φ↓ . ⊆ ⊆ 5 Combination: (simplified) d(φ) d(φ) d(ψ) (φ ψ)↓ = φ ψ↓ ∩ . ⊗ ⊗ 6 Domain: d(φ) φ↓ = φ. Marc Pouly An Introduction to Valuation Algebras 8/ 39 Valuation Algebras Local Computation Unifying Knowledge Representation Systems Semiring Induced Valuation Algebras Valuation Algebras Optimization as Inference Problem Valuation Algebra Instances Linear Predicate Manifolds Logic Indicator Semiring Functions Propositional Boolean Relations Algebras Logic Functions Distance Functions Valuation Algebras Gaussian Gaussian Belief Credal Hints Potentials Probability Functions Sets Potentials Possibility Constraint Potentials Systems Marc Pouly An Introduction to Valuation Algebras 9/ 39 Valuation Algebras The Inference Problem Local Computation Local Computation Semiring Induced Valuation Algebras A first Summary Optimization as Inference Problem Outline 1 Valuation Algebras 2 Local Computation The Inference Problem Local Computation A first Summary 3 Semiring Induced Valuation Algebras 4 Optimization as Inference Problem Marc Pouly An Introduction to Valuation Algebras 10/ 39 Valuation Algebras The Inference Problem Local Computation Local Computation Semiring Induced Valuation Algebras A first Summary Optimization as Inference Problem Inference Problem Query answering is the objective of knowledge processing. Task: Given a set of valuations and a query x, compute x (φ φ . φn)↓ 1 ⊗ 2 ⊗ ⊗ Query answering is a very natural process: 1 We combine all available knowledge. 2 The result is focused on the domain of interest. Marc Pouly An Introduction to Valuation Algebras 11/ 39 Valuation Algebras The Inference Problem Local Computation Local Computation Semiring Induced Valuation Algebras A first Summary Optimization as Inference Problem Inference Problems in Practice Probability Potentials: Compute marginal from joint probability distribution. ! Bayesian Network Evaluation Relations: Database query answering. Distance Functions: Shortest distance finding over multiple maps. Propositional Logic: If x = : Satisfiability Problem. ∅ Marc Pouly An Introduction to Valuation Algebras 12/ 39 Valuation Algebras The Inference Problem Local Computation Local Computation Semiring Induced Valuation Algebras A first Summary Optimization as Inference Problem Inference Problems in Practice Probability Potentials: Compute marginal from joint probability distribution. ! Bayesian Network Evaluation Relations: Database query answering. Distance Functions: Shortest distance finding over multiple maps. Propositional Logic: If x = : Satisfiability Problem. ∅ Marc Pouly An Introduction to Valuation Algebras 12/ 39 Valuation Algebras The Inference Problem Local Computation Local Computation Semiring Induced Valuation Algebras A first Summary Optimization as Inference Problem Inference Problems in Practice