  Domain adaptation (DA) arises as an important problem in statistical machine
learning when the source data used to train a model is different from the
target data used to test the model. Recent advances in DA have mainly been
application-driven and have largely relied on the idea of a common subspace for
source and target data. To understand the empirical successes and failures of
DA methods, we propose a theoretical framework via structural causal models
that enables analysis and comparison of the prediction performance of DA
methods. This framework also allows us to itemize the assumptions needed for
the DA methods to have a low target error. Additionally, with insights from our
theory, we propose a new DA method called CIRM that outperforms existing DA
methods when both the covariates and label distributions are perturbed in the
target data. We complement the theoretical analysis with extensive simulations
to show the necessity of the devised assumptions. Reproducible synthetic and
real data experiments are also provided to illustrate the strengths and
weaknesses of DA methods when parts of the assumptions in our theory are
violated.
