  Methods to identify cause-effect relationships currently mostly assume the
variables to be scalar random variables. However, in many fields the objects of
interest are vectors or groups of scalar variables. We present a new
constraint-based non-parametric approach for inferring the causal relationship
between two vector-valued random variables from observational data. Our method
employs sparsity estimates of directed and undirected graphs and is based on
two new principles for groupwise causal reasoning that we justify theoretically
in Pearl's graphical model-based causality framework. Our theoretical
considerations are complemented by two new causal discovery algorithms for
causal interactions between two random vectors which find the correct causal
direction reliably in simulations even if interactions are nonlinear. We
evaluate our methods empirically and compare them to other state-of-the-art
techniques.
