# Causal Reasoning with Ancestral Graphs

Jiji Zhang.

Year: 2008, Volume: 9, Issue: 47, Pages: 1437−1474

#### 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).