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...

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,

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 we first illustrate the above considerations for a variety of appealling criteria, and then, in an attempt to understand this behaviour, introduce a new game-theoretic framework for Probability Theory, the `prequential framework', which is particularly suited for the study of such problems.

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In this chapter, I review the main methods and techniques of complex systems science. As a

This article introduces a new wa of understanding subjective probabilit and its generalization to lower and upper prevision.

Some of the main users of statistical methods -- economists, social scientists, and epidemiologists -- are discovering that their fields rest not on statistical but on causal foundations. The blurring of these foundations over the years follows from the lack of mathematical notation capable of distinguishing causal from equational relationships. By providing formal and natural explication of such relations, graphical methods have the potential to revolutionize how statistics is used in knowledgerich applications. Statisticians, in response, are beginning to realize that causality is not a metaphysical deadend but a meaningful concept with clear mathematical underpinning. The paper surveys these developments and outlines future challenges.

This paper proposes a logic for causal based on event trees. Event trees provide a natural and familiar framework for probability and decision theory, but they lack the modularity that would make them convenient models for a logic. Our logic overcomes this deficiency by elaborating the idea of an event tree to that of an event space. This is a space containing situations or instantaneous events, which are related by relations of precedence, refinement, and impliation. 1