  We describe a formal approach to identify 'root causes' of outliers observed
in $n$ variables $X_1,\dots,X_n$ in a scenario where the causal relation
between the variables is a known directed acyclic graph (DAG). To this end, we
first introduce a systematic way to define outlier scores. Further, we
introduce the concept of 'conditional outlier score' which measures whether a
value of some variable is unexpected *given the value of its parents* in the
DAG, if one were to assume that the causal structure and the corresponding
conditional distributions are also valid for the anomaly. Finally, we quantify
to what extent the high outlier score of some target variable can be attributed
to outliers of its ancestors. This quantification is defined via Shapley values
from cooperative game theory.
