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...

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Introduction The introduction of Bayesian networks (Pearl 1986b) and associated local computation algorithms (Lauritzen and Spiegelhalter 1988, Shenoy and Shafer 1990, Jensen, Lauritzen and Olesen 1990) has initiated a renewed interest for understanding causal concepts in connection with modelling complex stochastic systems. It has become clear that graphical models, in particular those based upon directed acyclic graphs, have natural causal interpretations and thus form a base for a language in which causal concepts can be discussed and analysed in precise terms. As a consequence there has been an explosion of writings, not primarily within mainstream statistical literature, concerned with the exploitation of this language to clarify and extend causal concepts. Among these we mention in particular books by Spirtes, Glymour and Scheines (1993), Shafer (1996), and Pearl (2000) as well as the collection of papers in Glymour and Cooper (1999). Very briefly, but fundamentally,

The direct effect of one event on another can be defined and measured by holding constant all intermediate variables between the two. Indirect effects present conceptual and practical difficulties (in nonlinear models), because they cannot be isolated by holding certain variables constant. This paper presents a new way of defining the effect transmitted through a restricted set of paths, without controlling variables on the remaining paths. This permits the assessment of a more natural type of direct and indirect effects, one that is applicable in both linear and nonlinear models and that has broader policy-related interpretations. The paper establishes conditions under which such assessments can be estimated consistently from experimental and nonexperimental data, and thus extends path-analytic techniques to nonlinear and nonparametric models.

Structural equation modeling (SEM) has dominated causal analysis in the social and behavioral sciences since the 1960s. Currently, many SEM practitioners are having difficulty articulating the causal content of SEM and are seeking foundational answers.

This paper studies the causal interpretation of counterfactual sentences using a modifiable structural equation model. It is shown that two properties of counterfactuals, namely, composition and effectiveness, are sound and complete relative to this interpretation, when recursive (i.e., feedback-less) models are considered. Composition and effectiveness also hold in Lewis's closest-world semantics, which implies that for recursive models the causal interpretation imposes no restrictions beyond those embodied in Lewis's framework. A third property, called reversibility, holds in nonrecursive causal models but not in Lewis's closest-world semantics, which implies that Lewis's axioms do not capture some properties of systems with feedback. Causal inferences based on counterfactual analysis are exemplified and compared to those based on graphical models.

This paper summarizes basic concepts and principles that I have found to be useful in dealing with causal reasoning. The paper is written as a companion to a lecture under the same title, to be presented at IJCAI-99, and is intended to supplement the lecture with technical details and pointers to more elaborate discussions in the literature. The ruling conception will be to treat causation as a computational schema devised to identify the invariant relationships in the environment, so as to facilitate reliable prediction of the effect of actions. This conception, as well as several of its satellite principles and tools, has been guiding paradigm for several research communities in AI, most notably those connected with causal discovery, troubleshooting, planning under uncertainty and modeling the behavior of physical systems. My hopes are to encourage a broader and more effective usage of causal modeling by explicating these common principles in simple and familiar mathematical form. Af...

Chain graphs are a natural generalization of directed acyclic graphs (DAGs) and undirected graphs. However, the apparent simplicity of chain graphs belies the subtlety of the conditional independence hypotheses that they represent. There are a number of simple and apparently plausible, but ultimately fallacious interpretations of chain graphs that are often invoked, implicitly or explicitly. These interpretations also lead to awed methods for applying background knowledge to model selection. We present a valid interpretation by showing how the distribution corresponding to a chain graph may be generated as the equilibrium distribution of dynamic models with feedback. These dynamic interpretations lead to a simple theory of intervention, extending the theory developed for DAGs. Finally, we contrast chain graph models under this interpretation with simultaneous equation models which have traditionally been used to model feedback in econometrics. Keywords: Causal model; cha...

A probabilistic network is a graphical model that encodes probabilistic relationships between variables of interest. Such a model records qualitative influences between variables in addition to the numerical parameters of the probability distribution. As such it provides an ideal form for combining prior knowledge, which might be limited solely to experience of the influences between some of the variables of interest, and data. In this paper, we first show how data can be used to revise initial estimates of the parameters of a model. We then progress to showing how the structure of the model can be revised as data is obtained. Techniques for learning with incomplete data are also covered.

Consider a study whose design calls for the study subjects to be followed from enrollment (time t = 0) to time t = T,at which point a primary endpoint of interest Y is to be measured. The design of the study also calls for measurements on a vector V(t) of covariates to be made at one or more times t during the interval [0,T). We are interested in making inferences about the marginal mean µ0 of Y when some subjects drop out of the study at random times Q prior to the common fixed end of follow-up time T. The purpose of this article is to show how to make inferences about µ0 when the continuous drop-out time Q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables. In particular, we consider two models for the conditional hazard of drop-out given ( ¯ V(T), Y), where ¯ V(t) denotes the history of the process V(t) through time t, t ∈ [0,T). In the first model, we assume that λQ(t | ¯ V(T), Y) = λ0(t | ¯ V(t)) exp(α0Y), where α0 is a scalar parameter and λ0(t | ¯ V(t)) is an unrestricted positive function of t and the process ¯ V(t). When the process ¯ V(t) is high dimensional, estimation in this model is not feasible with moderate sample sizes, due to the curse of dimensionality. For such situations, we consider a second model that imposes the additional restriction that λ0(t | ¯ V(t)) = λ0(t) exp(γ ′ 0W(t)), where λ0(t) is an unspecified baseline hazard function, W(t) = w(t, ¯ V(t)), w(·, ·) is a known function that maps (t, ¯ V(t)) to Rq, and γ0 is a q × 1 unknown parameter vector. When α0 � = 0, then drop-out is nonignorable. On account of identifiability problems, joint estimation of the mean µ0 of Y and the selection bias parameter α0 may be difficult or impossible. Therefore, we propose regarding the selection bias parameter α0 as known, rather than estimating it from the data. We then perform a sensitivity analysis to see how inference about µ0 changes as we vary α0 over a plausible range of values. We apply our approach to the analysis of ACTG 175, an AIDS clinical trial. KEY WORDS: Augmented inverse probability of censoring weighted estimators; Cox proportional hazards model; Identification;