Algebras of Information. An Axiomatic Foundation

Abstract

The basic idea behind information algebras is that information comes in pieces, each referring to a certain question, that these pieces can be combined or aggregated and that the part relating to a given question can be extracted. This algebraic structure can be given different forms. Questions were originally represented by subsets of variables. Pieces of information were then represented by valuations associated with the domains of variables. This leads to an algebraic structure called valuation algebras. The basic axiomatics of this algebraic structure was in essence proposed by Shenoy and Shafer. Here a much more general view of systems of questions is proposed and pieces of information are related to the elements of this system of questions. This leads to a new and extended system of axioms for information algebras. Classical valuation algebras are essentially a special case of this new system. A full discussion of the algebraic theory of this new information algebras is given, including local computation, duality between labeled and domain-free versions of the algebras, order of information, finiteness of information and approximation, compact and continuous information algebras. Finally a rather complete discussion of uncertain information, based on random maps into information algebras is presented. This is shown to represent a generalisation of classical Dempster-Shafer theory.