  We assume that we have observational data generated from an unknown
underlying directed acyclic graph (DAG) model. A DAG is typically not
identifiable from observational data, but it is possible to consistently
estimate the equivalence class of a DAG. Moreover, for any given DAG, causal
effects can be estimated using intervention calculus. In this paper, we combine
these two parts. For each DAG in the estimated equivalence class, we use
intervention calculus to estimate the causal effects of the covariates on the
response. This yields a collection of estimated causal effects for each
covariate. We show that the distinct values in this set can be consistently
estimated by an algorithm that uses only local information of the graph. This
local approach is computationally fast and feasible in high-dimensional
problems. We propose to use summary measures of the set of possible causal
effects to determine variable importance. In particular, we use the minimum
absolute value of this set, since that is a lower bound on the size of the
causal effect. We demonstrate the merits of our methods in a simulation study
and on a data set about riboflavin production.
