Conditional products: An alternative approach to conditional independence

We introduce a new abstract approach to the study of conditional independence, founded on a concept analogous to the factoriza-tion properties of probabilistic independence, rather than the separation properties of a graph. The basic ingredient is the \con-ditional product", which provides a way of combining the objects under consideration while preserving as much independence as possible. We introduce an appropriate axiom system for conditional product, and show how, when these axioms are obeyed, it induces a derived concept of conditional independence which obeys the usual semi-graphoid axioms. The general structure is used to throw light on three speciic areas: the familiar probabilistic framework (both the discrete and the general case); a set-theoretic framework related to \variation independence"; and a variety of graphical frameworks. 1 MOTIVATION In several distinct mathematical areas, especially those describing uncertainty in Probability and Statistics some concept of conditional independence plays a fundamental role, permitting the decomposition of a complex object into simpler pieces. This concept was introduced in miscellaneous frameworks , but certain reasonable formal properties are shared. These formal properties are described by the abstract theory of conditional independence, as captured by the \semi-graphoid" axiom system (Dawid 1979; Spohn 1980; Pearl 1988). The archetypal model for this system is the concept of separation in an undi-rected graph, and the axioms can most readily be understood as being about a generalized form of separation. An entirely diierent approach to conditional independence is to try and abstract the factorization properties which form the traditional basis for the proba-bilistic deenition. This is the task we attempt here. We rst introduce the abstract concept of conditional product, and propose a suitable axiom system for it. We use this to introduce conditional independence as a derived concept, and show that it does then satisfy the semi-graphoid axioms. Such a point of view provides a unifying framework for conditional independence, and suggests new forms and applications. However, not every semi-graphoid can arise in this way. In this paper we describe the appropriate conditional product construction in probabilistic and set-theoretic frameworks for conditional independence, as well as showing that, in the frameworks of undirected or directed graphical models, no suitable conditional product exists. 2 CONDITIONAL PRODUCTS In this Section we give an axiomatic deenition of projection and conditional product. Then we show that an induced concept of conditional independence satis-es the semi-graphoid axioms.