Information usually comes in pieces, from different sources. It refers to different, but related questions. Therefore information needs to be aggregated and focused onto the relevant questions. Considering combination and focusing of information as the relevant operations leads to a generic algebraic structure for information. This book introduces and studies information from this algebraic point of view. Algebras of information provide the necessary abstract framework for generic inference procedures. They allow the application of these procedures to a large variety of different formalisms for representing information. At the same time they permit a generic study of conditional independence, a property considered as fundamental for knowledge presentation. Information algebras provide a natural framework to define and study uncertain information. Uncertain information is represented by random variables that naturally form information algebras. This theory also relates to probabilistic assumption-based reasoning in information systems and is the basis for the belief functions in the Dempster-Shafer theory of evidence.

Inhalt

1 Introduction.- 2 Valuation Algebras.- 2.1 The Framework.- 2.2 Axioms.- 2.3 Examples of Valuation Algebras.- 2.3.1 Indicator Functions.- 2.3.2 Relations.- 2.3.3 Probability Potentials.- 2.3.4 Possibility Potentials.- 2.3.5 Spohn Potentials.- 2.3.6 Set Potentials.- 2.3.7 Gaussian Potentials.- 2.4 Partial Marginalization.- 3 Algebraic Theory.- 3.1 Congruences.- 3.2 Domain-Free Valuation Algebras.- 3.3 Subalgebras, Homomorphisms.- 3.3.1 Subalgebras.- 3.3.2 Homomorphisms and Isomorphisms.- 3.3.3 Weak Subalgebras and Homomorphisms.- 3.4 Null Valuations.- 3.5 Regular Valuation Algebras.- 3.6 Separative Valuation Algebras.- 3.7 Scaled Valuation Algebras.- 4 Local Computation.- 4.1 Fusion Algorithm.- 4.2 Collect Algorithm.- 4.3 Computing Multiple Marginals.- 4.4 Architectures with Division.- 4.5 Computations in Valuation Algebras with Partial Marginalization.- 4.6 Scaling and Updating.- 5 Conditional Independence.- 5.1 Factorization and Graphical Models.- 5.2 Conditionals in Regular Algebras.- 5.3 Conditionals in Separative Algebras.- 6 Information Algebras.- 6.1 Idempotency.- 6.2 Partial Order of Information.- 6.3 File Systems.- 6.4 Information Systems.- 6.5 Examples.- 6.5.1 Propositional Logic.- 6.5.2 Boolean Information Algebras.- 6.5.3 Linear Equations.- 6.5.4 Linear Inequalities.- 6.6 Compact Systems.- 6.7 Mappings.- 7 Uncertain Information.- 7.1 Algebra of Random Variables.- 7.1.1 Random Variables.- 7.1.2 Allocation of Probability.- 7.1.3 Support Functions.- 7.2 Probabilistic Argumentation Systems.- 7.2.1 Assumption-Based Information.- 7.2.2 Probabilistic Argumentation Systems.- 7.2.3 Operations with Argumentation Systems.- 7.3 Allocations of Probability.- 7.3.1 Algebra of Allocations.- 7.3.2 Normalized Allocations.- 7.3.3 Random Variables and Allocations.- 7.3.4 Labeled Allocations and their Algebra.- 7.4 Independent Sources.- 7.4.1 Independent Random Variables.- 7.4.2 Algebra of Bpas.- References.