  Leveraging labelled data from multiple domains to enable prediction in
another domain without labels is a significant, yet challenging problem. To
address this problem, we introduce the framework DAPDAG (\textbf{D}omain
\textbf{A}daptation via \textbf{P}erturbed \textbf{DAG} Reconstruction) and
propose to learn an auto-encoder that undertakes inference on population
statistics given features and reconstructing a directed acyclic graph (DAG) as
an auxiliary task. The underlying DAG structure is assumed invariant among
observed variables whose conditional distributions are allowed to vary across
domains led by a latent environmental variable $E$. The encoder is designed to
serve as an inference device on $E$ while the decoder reconstructs each
observed variable conditioned on its graphical parents in the DAG and the
inferred $E$. We train the encoder and decoder jointly in an end-to-end manner
and conduct experiments on synthetic and real datasets with mixed variables.
Empirical results demonstrate that reconstructing the DAG benefits the
approximate inference. Furthermore, our approach can achieve competitive
performance against other benchmarks in prediction tasks, with better
adaptation ability, especially in the target domain significantly different
from the source domains.
