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Ordered Valuation Algebras a Generic Framework for Approximating Inference

Ordered Valuation Algebras a Generic Framework for Approximating Inference. Rolf Haenni UCLA Computer Science Department. Contents :. 1. Introduction 2. Valuation Algebras 3. Join Tree Propagation 4. Ordered Valuation Algebras 5. Resource-Bounded Approximation 6. Conclusion & Outlook.

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Ordered Valuation Algebras a Generic Framework for Approximating Inference

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  1. Ordered Valuation Algebras a Generic Framework for Approximating Inference Rolf Haenni UCLA Computer Science Department Contents: 1. Introduction 2. Valuation Algebras 3. Join Tree Propagation 4. Ordered Valuation Algebras 5. Resource-Bounded Approximation 6. Conclusion & Outlook

  2. 1. Introduction 1988: Local Computation (for probabilistic inference), Lauritzen & Spiegelhalter, J. Royal Statistical Society 1990: Valuations, Axioms, Propagation in Join TreesShenoy & Shafer, UAI’90 & Readings in Uncertain Reas. 1995: Fusion Algorithm, Binary Join Trees, Shenoy & Shafer,Tech. Rep., UAI’96, Int. J. of Approx. Reasoning‘97 1996: Bucket Elimination, Dechter, UAI’96, AI Journal ‘99 1997: Mini-Buckets, Dechter & Rish, IJCAI’97, Tech. Rep.’98 2000: Valuation & Information Algebras, Shenoy & Kohlas 2002: OrderedValuation Algebras, Haenni, AI Journal (?) 200?: Valuation Algebras, Kohlas, Comprehensive Book

  3. set of variables • valuation  “piece of information” • domain •  set of valuations with •  all valuations • knowledge base 2. Valuation Algebras Framework:

  4. labeling: • combination:  (aggregated information) • marginalization:  (information focussed to ) Problem of Inference: • compute  marginalize the joint valuation to the the domain of interest Operations:

  5. Axioms:

  6. Examples of Valuation Algebras: • probability potentials (CPT’s obtained from Bayesian network) • belief potentials (in the sense of Dempster-Shafer) • possibility functions • propositional logic • constraint systems • systems of linear equations and inequalities • relational algebra • etc. Valuation Algebra:  system satisfying (A1) to (A6)

  7. 3. Join Tree Porpagation Fusion Algorithm: (bucket elimination)  bucket

  8. Example:

  9. Join Tree BinaryJoin Tree  optimal structure for outward propagation LocalComputation

  10. Problems: • execution is often infeasible • effective running time is not predictable

  11. Completeness relation: is more complete than  is less complete than • information contained in is an approximationof the information contained in • representation of is more compact than representation of • sometimes, several completeness relationsexist (lower and upper approximation) 4. Ordered Valuation Algebras

  12. Additional axioms:

  13. Mini-Buckets:  not necessary for the following method Overview:

  14. with 5. Resource-Bounded Approximation Resource-bounded combination:

  15. share equally among the nodes of the join tree and redistribute unused portions Solution: Example: T = 100 s = 5 choose parameters during propagation (if the total time is restricted to ) Problem:

  16. estimatedtime neededat node n

  17. the result of the procedure is an approximation of the exact computation:  ,   Remarks: • the procedure stops after at most T milliseconds • method relies on the assumption that the time for marginalization is neglectable (but a similar procedure can be defined for cases where marginalization is expensive) • the same idea can be used for the outward propagation phase

  18. number of input parameters: < 20 infinitely many predictable running time: no yes no supports several approximations: yes no (but possible) proper axiomatic foundation: yes requires the existence of a new operator: no yes requires new axioms: yes yes no allows refining / anytime-algorithm: ??? “yes” for certain instanciations Comparison with mini-bucket elimination: Ordered Valuation Algebras MBE

  19. sec. Example: incomplete belief potentials  632 variables, 1101 valuations, 1100 combinations, max. |D|= 12

  20. 6. Conclusion & Outlook • Ordered valuation algebras provide a generic concept for approximating reasoning • A resource-bounded combination operator is the key for resource-bounded propagation algorithms (where the user determines the available computational resources). • Open questions are: • - the existence of a general refining procedure (which would lead to true anytime algorithms) • - how to treat cases where marginalization is more expensive than combination • - the range of its applicability

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