Title: E$^2$: Entropy Discrimination and Energy Optimization for Source-free Universal Domain Adaptation. Abstract: Universal domain adaptation (UniDA) aims to tackle the knowledge transfer problem in the presence of both distribution and category shifts. Most existing UniDA methods are developed based on the accessibility assumption of source-domain data during target model adaptation, which may result in privacy policy violation and source-data transfer inefficiency. To address this issue, we propose a novel source-free UniDA method by confidence-guided entropy discrimination and likelihood-induced energy optimization. The entropy-based separation criterion to determine known- and unknown-class target data may be too conservative for known-class prediction. Thus, we derive the confidence-guided entropy by scaling the normalized prediction score with the known-class confidence, such that much more known-class samples are correctly predicted. Without source-domain data for distribution alignment, we constrain the target-domain marginal distribution by maximizing the known-class likelihood and minimizing the unknown-class one. Since the marginal distribution is difficult to estimate but can be written as a function of free energy, the likelihood-induced loss is changed to an equivalent form based on energy optimization. Theoretically, the proposed method amounts to decreasing and increasing internal energy of known and unknown classes in physics, respectively. Extensive experiments on four publicly available datasets demonstrate the superiority of our method for source-free UniDA.