The primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject-matter information. In particular, the paper develops a principled, nonparametric framework for causal inference, in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data. If so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions; otherwise, the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained. Key words: Causal inference, graph models, interventions treatment effect 1 Introduction The tools introduced in this paper are aimed at helping researchers communicate qualitative assumptions about cause-effect relationships, elucidate the ramifications of such assumptions, and derive causal inferences from a combination...

We present a symbolic machinery that admits both probabilistic and causal information about a given domain, and produces probabilistic statements about the effect of actions and the impact of observations. The calculus admits two types of conditioning operators: ordinary Bayes conditioning, P (yjX = x), which represents the observation X = x, and causal conditioning, P (yjdo(X = x)), read: the probability of Y = y conditioned on holding X constant (at x) by deliberate action. Given a mixture of such observational and causal sentences, together with the topology of the causal graph, the calculus derives new conditional probabilities of both types, thus enabling one to quantify the effects of actions and observations. 1 Introduction Probabilistic methods, especially those based on graphical models have proven useful in tasks of predictions, abduction and belief revision [Pearl 1988, Heckerman 1990, Goldszmidt 1992, Darwiche 1993]. Their use in planning, however, remains less po...

This paper demonstrates the use of graphs as a mathematical tool for expressing independenices, and as a formal language for communicating and processing causal information in statistical analysis. We show how complex information about external interventions can be organized and represented graphically and, conversely, how the graphical representation can be used to facilitate quantitative predictions of the effects of interventions. We first review the Markovian account of causation and show that directed acyclic graphs (DAGs) offer an economical scheme for representing conditional independence assumptions and for deducing and displaying all the logical consequences of such assumptions. We then introduce the manipulative account of causation and show that any DAG defines a simple transformation which tells us how the probability distribution will change as a result of external interventions in the system. Using this transformation it is possible to quantify, from non-experimental data...

In this paper we study the identifiability of nonparametric models, that is, models in which both the functional forms of the equations and the probability distributions of the disturbances remain unspecified. Identifiability in such models does not mean uniqueness of parameters but rather uniqueness of the set of predictions of interest to the investigator. For example, predicting the effects of changes, interventions, and control. We provide sufficient and necessary conditions for identifying a set of causal predictions of the type: "Find the distribution of Y , assuming that X is controlled by external intervention", where Y and X are arbitrary variables of interest. Whenever identifiable, such predictions can be expressed in closed algebraic form, in terms of observed distributions. We also show how the identifying criteria can be verified qualitatively, by inspection, using the graphical representation of the structural model. When compared to standard identifiability tests of lin...

This paper highlights relationships among stochastic control theory, Lewis' notion of "imaging", and the representation of actions in AI systems. We show that the language of causal graphs offers a practical solution to the frame problem and its two satellites: the ramification and concurrency problems. Finally, we present a symbolic machinery that admits both probabilistic and causal information and produces probabilistic statements about the effect of actions and the impact of observations. 1 Representing and Revising Probability Functions Engineers consider the theory of stochastic control as the basic paradigm in the design and analysis of systems operating in uncertain environments. Knowledge in stochastic control theory is represented by a function P (s), which measures the probability assigned to each state s of the world, at any given time. Given P (s), it is possible to calculate the probability of any conceivable event E, by simply summing up P (s) over all states that entai...

Given an arbitrary causal graph, some of whose nodes are observable and some unobservable, the problem is to determine whether the causal effect of one variable on another can be computed from the joint distribution over the observables and, if the answer is positive, to derive a formula for the causal effect. We introduce a calculus which, using a step by step reduction of probabilistic expressions, derives the desired formulas. 1 1 Introduction Networks employing directed acyclic graphs (DAGs) can be used to provide either 1. an economical scheme for representing conditional independence assumptions and joint distribution functions, or 2. a graphical language for representing causal influences. Although the professed motivation for investigating such models lies primarily in the second category, [Wright, 1921, Blalock, 1971, Simon, 1954, Pearl 1988], causal inferences have been treated very cautiously in the statistical literature [Lauritzen & Spiegelhalter 1988, Cox 1992,...