Class imbalance remains a fundamental challenge in machine learning, where standard classifiers exhibit severe performance degradation in minority classes. Although existing approaches address imbalance through resampling or cost-sensitive learning during training, they require retraining or access to labeled target data when class distributions shift at deployment time, a common occurrence in real-world applications such as fraud detection, medical diagnosis, and anomaly detection. We present \textit{Online Bayesian Imbalanced Learning} (OBIL), a principled framework that decouples likelihood-ratio estimation from class-prior assumptions, enabling real-time adaptation to distribution shifts without model retraining. Our approach builds on the established connection between Bregman divergences and proper scoring rules to show that deep networks trained with such losses produce posterior probability estimates from which prior-invariant likelihood ratios can be extracted. We prove that these likelihood-ratio estimates remain valid under arbitrary changes in class priors and cost structures, requiring only a threshold adjustment for optimal Bayes decisions. We derive finite-sample regret bounds demonstrating that OBIL achieves $O(\sqrt{T \log T})$ regret against an oracle with perfect prior knowledge. Extensive experiments on benchmark datasets and medical diagnosis benchmarks under simulated deployment shifts demonstrate that OBIL maintains robust performance under severe distribution shifts, outperforming state-of-the-art methods in F1 Score when test distributions deviate significantly from the training conditions.