## Causal Reasoning with Ancestral Graphs

** Jiji Zhang**; 9(Jul):1437--1474, 2008.

### Abstract

Causal reasoning is primarily concerned with what would happen to a system under external interventions. In particular, we are often interested in predicting the probability distribution of some random variables that would result if some other variables were*forced*to take certain values. One prominent approach to tackling this problem is based on causal Bayesian networks, using directed acyclic graphs as

*causal*diagrams to relate post-intervention probabilities to pre-intervention probabilities that are estimable from observational data. However, such causal diagrams are seldom fully testable given observational data. In consequence, many causal discovery algorithms based on data-mining can only output an equivalence class of causal diagrams (rather than a single one). This paper is concerned with causal reasoning given an equivalence class of causal diagrams, represented by a (partial)

*ancestral graph*. We present two main results. The first result extends Pearl (1995)'s celebrated

*do-calculus*to the context of ancestral graphs. In the second result, we focus on a key component of Pearl's calculus---the property of

*invariance under interventions*, and give stronger graphical conditions for this property than those implied by the first result. The second result also improves the earlier, similar results due to Spirtes et al. (1993).

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