This paper develops axioms and formal semantics for statements of the form "X is causally irrelevant to Y in context Z," which we interpret to mean "Changing X will not affect Y if we hold Z constant." The axiomization of causal irrelevance is contrasted with the axiomization of informational irrelevance, as in "Learning X will not alter our belief in Y , once we know Z." Two versions of causal irrelevance are analyzed, probabilistic and deterministic. We show that, unless stability is assumed, the probabilistic definition yields a very loose structure, that is governed by just two trivial axioms. Under the stability assumption, probabilistic causal irrelevance is isomorphic to path interception in cyclic graphs. Under the deterministic definition, causal irrelevance complies with all of the axioms of path interception in cyclic graphs, with the exception of transitivity. We compare our formalism to that of [Lewis, 1973], and offer a graphical method of proving theorems abou...

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.

Abstract: From their inception, causal systems models (more commonly known as structural-equations models) have been accompanied by graphical representations or path diagrams that provide compact summaries of qualitative assumptions made by the models. These diagrams can be reinterpreted as probability models, enabling use of graph theory in probabilistic inference, and allowing easy deduction of independence conditions implied by the assumptions. They can also be used as a formal tool for causal inference, such as predicting the effects of external interventions. Given that the diagram is correct, one can see whether the causal effects of interest (target effects, or causal estimands) can be estimated from available data, or what additional observations are needed to validly estimate those effects. One can also see how to represent the effects as familiar standardized effect measures. The present article gives an overview of: (1) components of causal graph theory; (2) probability interpretations of graphical models; and (3) methodologic implications of the causal and probability structures encoded in the graph, such as sources of bias and the data needed for their control.

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, ...

We formulate the problem of learning and comparing the effects of dynamic treatment strategies in a probabilistic decision-theoretic framework, and in particular show how Robins’s “G-computation ” formula arises naturally. Careful attention is paid to the mathematical and substantive conditions necessary to justify use of this formula. Probabilistic influence diagrams are used to simplify manipulations. We compare our approach with formulations based on causal DAGs and on potential response models. Some key words and phrases: Causal inference; G-computation; Influence diagram; Observational study; Potential response; Sequential decision theory; Stability. 1

Abstract This paper builds on the labor econometrics and classical treatment effects literatures to provide a framework supporting causal concepts and methods for estimating effects of natural experiments operating over time in an explicitly dynamic time-series context. We examine conditions for the construction of covariates instrumental in identifying effects of interest that lead to new tests for unconfoundedness, a key condition for the identification of causal effects that we link to the concept of Granger non-causality. Our new tests for unconfoundedness are useful in both cross-section and dynamic time-series settings. Acknowledgments: The author is grateful for the comments and suggestions of the editor, two anony-

Directed acyclic graph (DAG) models are popular tools for describing causal relationships and for guiding attempts to learn them from data. In particular, they appear to supply a means of extracting causal conclusions from probabilistic conditional independence properties inferred from purely observational data. I take a critical look at this enterprise, and suggest that it is in need of more, and more explicit, methodological and philosophical justification than it typically receives. In particular, I argue for the value of a clean separation between formal causal language and intuitive causal assumptions.

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We consider probabilistic and graphical rules for detecting situations in which a dependence of one variable on another is altered by adjusting for a third variable (i.e., noncollapsibility), whether that dependence is causal or purely predictive. We focus on distinguishing situations in which adjustment will reduce, increase, or leave unchanged the degree of bias in an association of two variables when that association is taken to represent a causal effect of one variable on the other. We then consider situations in which adjustment may partially remove or introduce a potential source of bias in estimating causal effects, and some additional special cases useful for case-control studies, cohort studies with loss, and trials with noncompliance (nonadherence).

: This note is concerned with the class of hierarchical interaction models for mixed discrete and continuous variables as defined by Edwards (1990) and modified by Lauritzen (1996). In particular it is shown that any hierarchical log-linear interaction model can be generated by selection on a set of response variables in a directed Markov model over what we have termed the selection graph of the model. An inequality is established for the entries in the concentration matrix of any Gaussian undirected Markov distribution obtained by conditioning on the values of the response variables in the selection graph, thus demonstrating that not all such distributions can be generated in this way. Finally it is shown that in the mixed case only hierarchical models of the type defined by Edwards (1990) can be generated by selection as above. KEYWORDS: Bayesian networks; Conditional Gaussian distribution; Covariance selection; Gaussian graphical models; Log-linear interaction models; Recursive mode...