By Judea Pearl

This summarizes contemporary advances in causal inference and underscores the paradigmatic shifts that has to be undertaken in relocating from conventional statistical research to causal research of multivariate information. specified emphasis is put on the assumptions that underlie all causal inferences, the languages utilized in formulating these assumptions, the conditional nature of all causal and counterfactual claims, and the tools which were built for the overview of such claims. those advances are illustrated utilizing a basic concept of causation in response to the Structural Causal version (SCM), which subsumes and unifies different ways to causation, and offers a coherent mathematical starting place for the research of factors and counterfactuals. particularly, the paper surveys the improvement of mathematical instruments for inferring (from a mixture of information and assumptions) solutions to 3 different types of causal queries: these approximately (1) the consequences of capability interventions, (2) chances of counterfactuals, and (3) direct and oblique results (also often called "mediation"). eventually, the paper defines the formal and conceptual relationships among the structural and potential-outcome frameworks and provides instruments for a symbiotic research that makes use of the powerful beneficial properties of either. The instruments are established within the analyses of mediation, motives of results, and possibilities of causation.

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**Extra resources for An Introduction to Causal Inference**

**Example text**

View larger versionFigure 4: Markovian model illustrating the back-door criterion. Error terms are not shown explicitly. The following criterion, named “back-door” in (Pearl, 1993a), settles this problem by providing a graphical method of selecting admissible sets of factors for adjustment. Definition 3 (Admissible sets – the back-door criterion) A set S is admissible (or “sufficient”) for adjustment if two conditions hold: No element of S is a descendant of X The elements of S “block” all “back-door” paths from X to Y, namely all paths that end with an arrow pointing to X.

Formally, the implication of finding an admissible set S is that, stratifying on S is guaranteed to remove all confounding bias relative the causal effect of X on Y. In other words, the risk difference in each stratum of S gives the correct causal effect in that stratum. In the binary case, for example, the risk difference in stratum s of S is given by P(Y = 1|X = 1, S = s)−P(Y = 1|X = 0, S = s) while the causal effect (of X on Y) at that stratum is given by P(Y = 1|do(X = 1), S = s)−P(Y = 1|do(X = 0), S = s).

Clearly, the distribution P(uY, uX, uZ) induces a well defined probability on the counterfactual event Yx0 = y, as well as on joint counterfactual events, such as ‘Yx0 = y AND Yx1 = y′,’ which are, in principle, unobservable if x0 ≠ x1. Thus, to answer attributional questions, such as whether Y would be y1 if X were x1, given that in fact Y is y0 and X is x0, we need to compute the conditional probability P(Yx1 = y1|Y = y0, X = x0) which is well defined once we know the forms of the structural equations and the distribution of the exogenous variables in the model.

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