  The establishment of the link between causality and unsupervised domain
adaptation (UDA)/semi-supervised learning (SSL) has led to methodological
advances in these learning problems in recent years. However, a formal theory
that explains the role of causality in the generalization performance of
UDA/SSL is still lacking. In this paper, we consider the UDA/SSL setting where
we access m labeled source data and n unlabeled target data as training
instances under a parametric probabilistic model. We study the learning
performance (e.g., excess risk) of prediction in the target domain.
Specifically, we distinguish two scenarios: the learning problem is called
causal learning if the feature is the cause and the label is the effect, and is
called anti-causal learning otherwise. We show that in causal learning, the
excess risk depends on the size of the source sample at a rate of O(1/m) only
if the labelling distribution between the source and target domains remains
unchanged. In anti-causal learning, we show that the unlabeled data dominate
the performance at a rate of typically O(1/n). Our analysis is based on the
notion of potential outcome random variables and information theory. These
results bring out the relationship between the data sample size and the
hardness of the learning problem with different causal mechanisms.
