this paper, we propose an automated process for constructing the combined dependency structure of a ########## probabilistic network. Each domain expert supplies any known conditional independency information and not necessarily an explicit dependency structure. Our method determines a succinct representation of all the supplied independency information called a ####### #####. This process involves detecting all ############ information and removing all ######### information. A ###### dependency structure of the multiagent probabilistic network can be constructed directly from this minimal cover. The main result of this paper is that the constructed dependency structure is a ########### of the minimal cover. That is, every probabilistic conditional independency logically implied by the minimal cover can be inferred from the dependency structure and every probabilistic conditional independency inferred from the dependency structure is logically implied by the minimal cover

This paper investigates Walley’s concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can violate the symmetry, contraction and intersection axioms, that epistemic independence can violate contraction and intersection, and that this accords with informal notions of irrelevance and independence.

. Given a collection of random variables [¸ i ] i2N where N is a finite nonempty set, the corresponding multiinformation function ascribes the relative entropy of the joint distribution of [¸ i ] i2A with respect to the product of distributions of individual random variables ¸ i for i 2 A to every subset A ae N . We argue it is a useful tool for problems concerning stochastic (conditional) dependence and independence (at least in discrete case). First, it makes possible to express the conditional mutual information between [¸ i ] i2A and [¸ i ] i2B given [¸ i ] i2C (for every disjoint A; B; C ae N) which can be considered as a good measure of conditional stochastic dependence. Second, one can introduce reasonable measures of dependence of level r among variables [¸ i ] i2A (where A ae N , 1 r ! card A) which are expressible by means of the multiinformation function. Third, it enables one to derive theoretical results on (nonexistence of an) axiomatic characterization of stochastic c...

This article has been prepared as an entry for the Wiley Encyclopedia of Statistical Sciences (Update). It gives a brief overview of fundamental properties and applications of conditional independence. ESS Update A. P. Dawid Conditional Independence Ancillarity; axioms; graphical models; markov properties; sufficiency. The concepts of independence and conditional independence (CI) between random variables originate in Probability Theory, where they are introduced as properties of an underlying probability measure P on the sample space (see CONDITIONAL PROBABILITY AND EXPECTATION). Much of traditional Probability Theory and Statistics involves analysis of distributions having such properties: for example, limit theorems for independent and identically distributed variables, or the theory of MARKOV PROCESSES. More recently, it has become apparent that it is fruitful to treat conditional independence (and its special case independence) as a primitive concept, with an intuitive meaning, ...

this paper, the semigraphoid closure of every couple of CI-statements is proved to be a CI-model. The substantial step to it is to show that every probabilistically sound inference rule for axiomatic characterization of CI properties (= axiom), having at most two antecedents, is a consequence of the semigraphoid inference rules. Moreover, all potential dominant triplets of the mentioned semigraphoid closure are found.

. Special conditional independence structures have been recognized to be matroids. This opens new possibilities for the application of matroid theory methods (duality, minors, expansions) to the study of conditional independence and, on the other hand, starts a new probabilistic branch of matroid representation theory. 2 1. INTRODUCTION Though of old origin, the concept of conditional stochastic independence has been reattracting attention of mathematicians during last decades. A new viewpoint, from Dawid (1979), consists in the simultaneous examination of all conditional independences (among triples of subsystems of a stochastic system) separately from the joint probability distribution. Subsequent development of this idea has been influenced by the logic of integrity constraints from databases (see Pearl(1988), Geiger, Pearl(1989), Studen'y(1992)) and is oriented toward searching for plausible conditional independence relations (Oliver, Smith(1990), Mat'us(1992a)). In the thirties s...

In this paper the concept of conditional independence (CI) within the framework of natural conditional functions (NCF) is studied. An NCF is a function ascribing natural numbers to possible states of the world; it is the central concept of Spohn's theory of deterministic epistemology. Basic properties of CI within this framework are recalled and further results analogical to the results concerning stochastic CI are proved. Firstly, the intersection of two CI--models is shown to be a CI--model. Using this it is proved that CI--models for NCFs have no finite complete axiomatic characterization (by means of a simple deductive system describing relationships among CI--statements). The last part is devoted to the marginal problem for NCFs where it is shown that the (pairwise) consonancy is equivalent to the consistency iff the running intersection property holds. KEYWORDS: natural conditional function, conditional independence, axiomatic characterization, marginal problem, running intersect...

We explore the conditional probabilistic independences of systems of random variables (I ; J jK), to read "I is conditionally independent of J given K", which are ascending (I ; J jK) ) (I ; J jK [ L) or descending (I ; J jK [ L) ) (I ; J jK). The resulting abstract independence structures can be equivalently described by weak families of connected sets. Using, in addition, probabilistic representations of matroids we present probabilistic representations of the structures and formulate duality conjectures. Local and global forms of semigraphoids and these independence structures are elaborated and special cases including unconditional independence relations, fixed--context relations and Markov networks are discussed. 1 INTRODUCTION Let N be a fixed finite set, P(N) the family of its subsets, R(N) the family of all triples (i; jjK), where K ae N; i; j 2 N \Gamma K; i 6= j, and T (N) the family of all triples (I ; J jK), where I ; J; K ae N disjoint. We shall not distinguish between ...

Researchers in artificial intelligence and decision analysis share a concern with the construction of formal models of human knowledge and expertise. Historically, however, their approaches to these problems have diverged. Members of these two communities have recently discovered common ground: a family of graphical models of decision theory known as influence diagrams or as belief networks. These models are equally attractive to theoreticians, decision modelers, and designers of knowledge-based systems. From a theoretical perspective, they combine graph theory, probability theory and decision theory. From an implementation perspective, they lead to powerful automated systems. Although many practicing decision analysts have already adopted influence diagrams as modeling and structuring tools, they may remain unaware of the theoretical work that has emerged from the artificial intelligence community. This paper surveys the first decade or so of this work. Investment Technology Group, ...

In this article, we consider the computational aspects of deciding whether a conditional independence statement t is implied by a list of conditional independence statements L using the independence implication provided by the method of structural imsets. We present two algorithmic methods which have the interesting complementary properties that one method performs well to prove that t is implied by L, while the other performs well to prove that t is not implied by L. However, both methods do not well perform the opposite. This gives rise to a parallel algorithm in which both methods race against each other in order to determine effectively whether t is or is not implied. Some empirical evidence is provided that suggests this racing algorithms method performs considerably better than an existing method based on so-called skeletal characterization of the respective implication. Furthermore, unlike previous methods, the method is able to handle more than five variables.