In this work, we propose an adversarial unsupervised domain adaptation (UDA) approach with the inherent conditional and label shifts, in which we aim to align the distributions w.r.t. both $p(x|y)$ and $p(y)$. Since the label is inaccessible in the target domain, the conventional adversarial UDA assumes $p(y)$ is invariant across domains, and relies on aligning $p(x)$ as an alternative to the $p(x|y)$ alignment. To address this, we provide a thorough theoretical and empirical analysis of the conventional adversarial UDA methods under both conditional and label shifts, and propose a novel and practical alternative optimization scheme for adversarial UDA. Specifically, we infer the marginal $p(y)$ and align $p(x|y)$ iteratively in the training, and precisely align the posterior $p(y|x)$ in testing. Our experimental results demonstrate its effectiveness on both classification and segmentation UDA, and partial UDA.
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