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...

tion of belief networks is given in [4]. 2 In [3], the graphs were called "causal networks," for which the authors were criticised; they have agreed to refrain from using the word "causal." In the current paper, Spiegelhalter etal. deemphasize the causal interpretation of the arcs in favor of the "irrelevance" interpretation (page 4). I think this retreat is regrettable for two reasons: first, causal associations are the primary source of judgments about irrelevance and, second, rejecting the causal interpretation of arcs prevents us from using graphical models for making legitimate predictions about the effect of actions. Such predictions are indispensable in applications such as treatment management and patient monitoring. the causal model also tells us how these probabilities would change as a result of external interventions in the system. For this reason, causal models (or "structural models" as they are often called) have been the target of relent

Evaluation of counterfactual queries (e.g., "If A were true, would C have been true?") is important to fault diagnosis, planning, and determination of liability. In this paper we present methods for computing the probabilities of such queries using the formulation proposed in [Balke and Pearl, 1994], where the antecedent of the query is interpreted as an external action that forces the proposition A to be true. When a prior probability is available on the causal mechanisms governing the domain, counterfactual probabilities can be evaluated precisely. However, when causal knowledge is specified as conditional probabilities on the observables, only bounds can computed. This paper develops techniques for evaluating these bounds, and demonstrates their use in two applications: (1) the determination of treatment efficacy from studies in which subjects may choose their own treatment, and (2) the determination of liability in product-safety litigation. 1 INTRODUCTION A counterfactual sente...

This paper does not attempt to correct for shortcomings of the traditional IV method but, rather, to develop a complementary method which can provide unbiased estimates under conditions where the IV method fails. The method relies on finding an auxiliary variable Z

Experimental studies in which treatment assignment is random but subject compliance is imperfect may be susceptible to bias; the actual effect of the treatment may deviate appreciably from the mean difference between treated and untreated subjects. This paper establishes universal formulas that can be used to bound the actual treatment effect in any experiment for which compliance data is available and in which the assignment influences the response only through the treatment given. Using a linear programming analysis, we present formulas that provide the tightest bounds that can be inferred on the average treatment effect, given an empirical distribution of assignments, treatments, and responses. The application of these results is demonstrated on data that relates cholesterol levels to cholestyramine treatment ([Lipid Research Clinic Program 84]).

The situation calculus is a popular technique for reasoning about action and change. However, its restriction to a firstorder syntax and pure deductive reasoning makes it unsuitable in many contexts. In particular, we often face uncertainty, due either to lack of knowledge or to some probabilistic aspects of the world. While attempts have been made to address aspects of this problem, most notably using nonmonotonic reasoning formalisms, the general problem of uncertainty in reasoning about action has not been fully dealt with in a logical framework. In this paper we present a theory of action that extends the situation calculus to deal with uncertainty. Our framework is based on applying the random-worlds approach of [BGHK94] to a situation calculus ontology, enriched to allow the expression of probabilistic action effects. Our approach is able to solve many of the problems imposed by incomplete and probabilistic knowledge within a unified framework. In particular, we obtain a default ...

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Introduction It is common in the literature on instrumental variables to remark upon the difficulty of knowing or demonstrating that a potential instrument is exogenous, in the sense of being uncorrelated with the disturbances [Bartels, 1991, Johnston, 1972]. It is also widely recognized that exogeneity is an assumption embedded in the model specification [Engle, et al, 1984], hence, it rests on subjective judgment and, like other structural assumptions of causation and "zero-restrictions", it cannot be tested in purely observational studies. The purpose of this note is to show that despite its elusive nature, exogeneity can nevertheless be given some empirical test. The test is not guaranteed to detect all violations of exogeneity but it can, in certain circumstances, screen away real bad choices of would-be instruments. 2 An Instrumental Inequality Definition 2.1 (exogeneity) A variable z is said to be exogenous relative to an ordered

Principal stratification has recently become a popular tool to address certain causal inference questions particularly in dealing with post-randomization factors in randomized trials. Here we analyze the conceptual basis for this framework and invite response to clarify the value of principal stratification in estimating causal effects of interest.

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...