  Multi-source domain adaptation (MSDA) addresses the challenge of learning a
label prediction function for an unlabeled target domain by leveraging both the
labeled data from multiple source domains and the unlabeled data from the
target domain. Conventional MSDA approaches often rely on covariate shift or
conditional shift paradigms, which assume a consistent label distribution
across domains. However, this assumption proves limiting in practical scenarios
where label distributions do vary across domains, diminishing its applicability
in real-world settings. For example, animals from different regions exhibit
diverse characteristics due to varying diets and genetics.
  Motivated by this, we propose a novel paradigm called latent covariate shift
(LCS), which introduces significantly greater variability and adaptability
across domains. Notably, it provides a theoretical assurance for recovering the
latent cause of the label variable, which we refer to as the latent content
variable. Within this new paradigm, we present an intricate causal generative
model by introducing latent noises across domains, along with a latent content
variable and a latent style variable to achieve more nuanced rendering of
observational data. We demonstrate that the latent content variable can be
identified up to block identifiability due to its versatile yet distinct causal
structure. We anchor our theoretical insights into a novel MSDA method, which
learns the label distribution conditioned on the identifiable latent content
variable, thereby accommodating more substantial distribution shifts. The
proposed approach showcases exceptional performance and efficacy on both
simulated and real-world datasets.
