  Previous studies have shown that leveraging domain index can significantly
boost domain adaptation performance (arXiv:2007.01807, arXiv:2202.03628).
However, such domain indices are not always available. To address this
challenge, we first provide a formal definition of domain index from the
probabilistic perspective, and then propose an adversarial variational Bayesian
framework that infers domain indices from multi-domain data, thereby providing
additional insight on domain relations and improving domain adaptation
performance. Our theoretical analysis shows that our adversarial variational
Bayesian framework finds the optimal domain index at equilibrium. Empirical
results on both synthetic and real data verify that our model can produce
interpretable domain indices which enable us to achieve superior performance
compared to state-of-the-art domain adaptation methods. Code is available at
https://github.com/Wang-ML-Lab/VDI.
