  We consider the problem of causal structure learning in the setting of
heterogeneous populations, i.e., populations in which a single causal structure
does not adequately represent all population members, as is common in
biological and social sciences. To this end, we introduce a distance
covariance-based kernel designed specifically to measure the similarity between
the underlying nonlinear causal structures of different samples. Indeed, we
prove that the corresponding feature map is a statistically consistent
estimator of nonlinear independence structure, rendering the kernel itself a
statistical test for the hypothesis that sets of samples come from different
generating causal structures. Even stronger, we prove that the kernel space is
isometric to the space of causal ancestral graphs, so that distance between
samples in the kernel space is guaranteed to correspond to distance between
their generating causal structures. This kernel thus enables us to perform
clustering to identify the homogeneous subpopulations, for which we can then
learn causal structures using existing methods. Though we focus on the
theoretical aspects of the kernel, we also evaluate its performance on
synthetic data and demonstrate its use on a real gene expression data set.
