Generic Inference

This book provides a rigorous algebraic study of the most popular inference formalisms with a special focus on their wide application area, showing that all these tasks can be performed by a single generic inference algorithm. Written by the leading international authority on the topic, it includes an algebraic perspective (study of the valuation algebra framework), an algorithmic perspective (study of the generic inference schemes) and a "practical" perspective (formalisms and applications). Researchers in a number of fields including artificial intelligence, operational research, databases and other areas of computer science; graduate students; and professional programmers of inference methods will benefit from this work. List of Instances and Applications.

List of Figures and Tables.

Acknowledgments.

Introduction.

Part I Local Computation.

1 Valuation Algebras.

1.1 Operations and Axioms.

1.2 First Examples.

1.3 Conclusion.

2 Inference Problems.

2.1 Graphs, Trees and Hypergraphs.

2.2 Knowledgebases and their Representation.

2.3 The Inference Probloem.

2.4 Conclusion.

3 Computing Single Queries.

3.1 Valuation Algebras with Variable Elimination.

3.2 Fusion and Bucker Elimination.

3.3 Valuation Algebras with Neutral Elements.

3.4 Valuation Algebras with Null Elements.

3.5 Local Computation as Message Passing Scheme.

3.6 Covering Join Trees.

3.7 Join Tree Construction.

3.8 The Collect Algorithm.

3.9 Adjoining an Identity Element.

3.10 The Generalized Collect Algorithm.

3.11 An Application: The Fast Fourier Transform.

3.12 Conclusion.

4 Computing Multiple Queries.

4.1 The Shenoy Shafer.

4.2 Valuation Algebras with Inverse Elements.

4.3 The Lauritzen Spiegelhalter Architecture.

4.4 The HUGIN Architecture.

4.5 The Idempotent Architecture.

4.6 Answering Uncovered Queries.

4.7 Scaling and Normalization.

4.8 Local Computation with Scaling.

4.9 Conclusion.

Part II Generic Constructions.

5 Semiring Valuation Algebras.

5.1 Semirings.

5.2 Semirings and Order.

5.3 Semiring Valuation Algebras.

5.4 Examples of Semiring Valuation Algebras.

5.5 Properties of Semiring Valuation Algebras.

5.6 Some Computational Aspects.

5.7 Set Based Semiring Valuation Algebras.

5.8 Properties of Set Based Semiring Valuation Algebras.

5.9 Conclusion.

6 Valuation Algebras for Path Problems.

6.1 Some Path Problem Examples.

6.2 The Algebraic Path Problem.

6.3 Quasi Regular Semirings.

6.4 Quasi Regular Valuation Algebras.

6.5 Properties of Quasi Regular Valuation Algebras.

6.6 Kleene Algebras.

6.7 Kleene Valuation Algebras.

6.8 Properties of Kleene Valuation Algebras.

6.9 Further Path Problems.

6.10 Conclusion.

7 Language and Information.

7.1 Propositional Logic.

7.2 Linear Equations.

7.3 Information in Context.

7.4 Conclusion.

Part III Applications.

8 Dynamic Programming.

8.1 Solutions and Solution Extensions.

8.2 Computing Solutions.

8.3 Optimization and Constraint Problems.

8.4 Computing Solutions of Optimization Problems.

8.5 Conclusion.

9 Sparse Matrix Techniques.

9.1 Systems of Linear Equations.

9.2 Symmetric, Positive Definite Matrices.

9.3 Semiring Fixpoint Equation Systems.

9.4 Conclusion.

10 Gaussian Information.

10.1 Gaussian Systems and Potentials.

10.2 Generalized Gaussian Potentials.

10.3 Gaussian Information and Gaussian Potentials.

10.4 Valuation Algebra of Gaussian Potentials.

10.5 An Application: Gaussian Dynamic Systems.

10.6 An Application: Gaussian Bayesian Networks.

10.7 Conclusion.

Appendix.

References.

Index.

Marc Pouly, PhD, received the Award for Outstanding PhD Thesis in Computer Science at the University of Fribourg (Switzerland), in 2008. He was visiting researcher at the Cork Constraint Computation Centre in Ireland and, since 2010, he is researcher at the Interdisciplinary Centre for Security, Reliability and Trust of the University of Luxembourg.

Jürg Kohlas, PhD, is Professor of Theoretical Computer Science in the Department of Informatics at the University of Fribourg (Switzerland). His research interests include algebraic theory of information and probabilistic argumentation.

A Rigorous Algebraic Study of the Most Popular Inference Formalisms

This unique text provides a complete algebraic and algorithmic study of generic inference methods that are derived from the general valuation algebra framework, with special focus on the many practical applications in computer science. Written by the leading international authorities on the topic, Generic Inference is divided into three parts:

• Part I defines the valuation algebra framework and gives a first catalog of practically important examples; explains the generic inference problem and surveys fundamental applications that require the solution of such problems with knowledge bases from different valuation algebras; develops generic algorithms for the solution of single- and multiple-query inference problems with arbitrary valuation algebras; and discusses issues related to complexity and optimization

• Part II identifies several important families of valuation algebras derived from other mathematical structures—including soft constraints, path problems, linear systems, and logical structures—and uncovers the close relationship between valuation algebras and semiring theory

• Part III discusses various applications of generic inference, with chapters dedicated to dynamic optimization, sparse matrix techniques, and linear systems with stochastic disturbances

The text is accompanied by a large number of examples; at the end of every chapter are selected exercises and open research problems. A comprehensive bibliography on valuation algebras and local computation is provided, and all algorithms are developed mathematically and given in pseudo-code. Generic Inference is designed for researchers in a number of fields, including artificial intelligence, operational research, databases, and other areas of computer science; graduate students and other researchers interested in general reasoning frameworks; and professional programmers of inference methods.