  Understanding causal relationships is one of the most important goals of
modern science. So far, the causal inference literature has focused almost
exclusively on outcomes coming from a linear space, most commonly the Euclidean
space. However, it is increasingly common that complex datasets collected
through electronic sources, such as wearable devices and medical imaging,
cannot be represented as data points from linear spaces. In this paper, we
present a formal definition of causal effects for outcomes from non-linear
spaces, with a focus on the Wasserstein space of cumulative distribution
functions. We develop doubly robust estimators and associated asymptotic theory
for these causal effects. Our framework extends to outcomes from certain
Riemannian manifolds. As an illustration, we use our framework to quantify the
causal effect of marriage on physical activity patterns using wearable device
data collected through the National Health and Nutrition Examination Survey.
