Thomas Verma

A theory of inferred causation

Publisher Summary This chapter discusses the theory of inferred causation. The study of causation is central to the understanding of human reasoning. Inferences involving changing environments require causal theories that make formal distinctions between beliefs based on passive observations and those reflecting intervening actions. In applications such as diagnosis, qualitative physics, and plan recognition, a central task is that of finding a satisfactory explanation to a given set of observations, and the meaning of explanation is intimately related to the notion of causation. In some systems, causal ordering is defined as the ordering at which subsets of variables can be solved independently of others; in other systems, it follows the way a disturbance is propagated from one variable to others. An empirical semantics for causation is important for several reasons. The notion of causation is often associated with those of necessity and functional dependence; causal expressions often tolerate exceptions, primarily because of missing variables and coarse description. Temporal precedence is normally assumed essential for defining causation, and it is one of the most important clues that people use to distinguish causal from other types of associations.

Identifying independence in bayesian networks

An important feature of Bayesian networks is that they facilitate explicit encoding of information about independencies in the domain, information that is indispensable for efficient inferencing. This article characterizes all independence assertions that logically follow from the topology of a network and develops a linear time algorithm that identifies these assertions. The algorithms correctness is based on the soundness of a graphical criterion, called d-separation, and its optimality stems from the completeness of d-separation. An enhanced version of d-separation, called D-separation, is defined, extending the algorithm to networks that encode functional dependencies. Finally, the algorithm is shown to work for a broad class of nonprobabilistic independencies.

An algorithm for deciding if a set of observed independencies has a causal explanation

An Algorithm for Deciding if a Set of Observed Has a Causal Explanation Thomas Verma Northrop Corporation One Research Park Palos Verdes, CA 90274 verma@nrtc.northrop.com Abstract In a previous paper [Pearl and Verma, 1991] we presented an algorithm for extracting causal influences from independence informa- tion, where a causal influence was defined as the existence of a directed arc in all mini- mal causal models consistent with the data. In this paper we address the question of de- ciding whether there exists a causal model that explains ALL the observed dependencies and independencies. Formally, given a list M of conditional independence statements, it is required to decide whether there exists a directed acyclic graph (dag) D that is per- fectly consistent with M, namely, every state- ment in M, and no other, is reflected via d- separation in D. We present and analyze an effective algorithm that tests for the existence of such a dag, and produces one, if it exists. Introduction Directed acyclic graphs (dags) have been widely used for modeling statistical data. Starting with the pio- neering work of Sewal Wright [Wright, 1921] who in- troduced path analysis to statistics, through the more recent development of Bayesian networks and influence diagrams, dag structures have served primarily for en- coding causal influences between variables as well as between actions and variables. Even statisticians who usually treat causality with ex- treme caution, have found the structure of dags to be an advantageous model for explanatory purposes. N. Wermuth, for example, mentions several such ad- vantages [Wermuth, 1991]. First, the dag describes a stepwise stochastic process by which the data could have been generated and in this sense it may even prove the basis for developing causal explanations [Cox, 1992]. Second, each parameter in the dag has a well understood meaning since it is a conditional probability, i.e., it measures the probability of the re- sponse variable given a particular configuration of the Independencies Judea Pearl Cognitive Systems Laboratory Computer Science Department University of California Los Angeles, CA 90024 judea@cs.ucla.edu explanatory (parents) variables and all other variables being unspecified. Third, the task of estimating the parameters in the dag model can be decomposed into a sequence of local estimation analyses, each involv- ing a variable and its parent set in the dag. Fourth, general results are available for reading all implied independencies directly off the dag [Verma, 1986], [Pearl, 1988], [Lauritzen et al., 1990] and for deciding from the topology of two given dags whether they are equivalent, i.e., whether they specify the same set of independence-restrictions on the joint distribu- tion [Frydenberg, 1990], [Verma and Pearl, 1990], and whether one dag specifies more restrictions than the other [Pearl et al., 1989] . This paper adds a fifth advantage to the list above. It presents an algorithm which decides for an ar- bitrary list of conditional independence statements whether it defines a dag and, if it does, a correspond- ing dag is drawn. The algorithm we present has its basis in the Inferred-Causation (IC) algorithm de- scribed in [Pearl and Verma, 1991] and in Lemmas 1 and 2 of [Verma and Pearl, 1990]. Whereas in [Pearl and Verma, 1991] we were interested in detect- ing local relationships that we called genuine causal influences , we now consider an entire dag as one unit which ought to fit the data at hand. Problem Given a list M of conditional independence statements ranging over a set of variables U it is required to decide whether there exists a directed acyclic graph (dag) D that is consistent with M. Our analysis will focus on lists that are closed un- der the graphoid axioms (see Appendix for definition). Section 5 will discuss possible extensions to lists which are not closed. The criterion for dag equivalence is given in Corol- lary 3.2. It follows from Frydenbergs analysis of chain graphs, which applies to strictly positive distribu- tions. The more direct analysis of Verma and Pearl [Verma and Pearl, 1990] renders the criterion applicable to arbitrary distributions, as well as to non-probabilistic de- pendencies of the graphoid type [Pearl and Paz, 1986].

d-Separation: From Theorems to Algorithms

An efficient algorithm is developed that identifies all independencies implied by the topology of a Bayesian network. Its correctness and maximality stems from the soundness and completeness of d -separation with respect to probability theory. The algorithm runs in time 0 (|E|) where E is the number of edges in the network

A statistical semantics for causation

We propose a model-theoretic definition of causation, and show that, contrary to common folklore, genuine causal influences can be distinguished from spurious covariations following standard norms of inductive reasoning. We also establish a sound characterization of the conditions under which such a distinction is possible. Finally, we provide a proof-theoretical procedure for inductive causation and show that, for a large class of data and structures, effective algorithms exist that uncover the direction of causal influences as defined above.

Deciding morality of graphs is NP-complete

In order to find a causal explanation for data presented in the form of covariance and concentration matrices it is necessary to decide if the graph formed by such associations is a projection of a directed acyclic graph (dag). We show that the general problem of deciding whether such a dag exists is NP-complete.

Causal Networks: Semantics and Expressiveness* *This work was partially supported by the National Science Foundation Grant #IRI-8610155. “Graphoids: A Computer Representation for Dependencies and Relevance in Automated Reasoning (Computer Information Science).”

Dependency knowledge of the form “ x is independent of y once z is known” invariably obeys the four graphoid axioms, examples include probabilistic and database dependencies. Often, such knowledge can be represented efficiently with graphical structures such as undirected graphs and directed acyclic graphs (DAGs). In this paper we show that the graphical criterion called d-separation is a sound rule for reading independencies from any DAG based on a causal input list drawn from a graphoid. The rule may be extended to cover DAGs that represent functional dependencies as well as conditional dependencies.