Effective adaptation to distribution shifts in training data is pivotal for sustaining robustness in neural networks, especially when removing specific biases or outdated information, a process known as machine unlearning. Traditional approaches typically assume that data variations are random, which makes it difficult to adjust the model parameters accurately to remove patterns and characteristics from unlearned data. In this work, we present Unlearning Information Bottleneck (UIB), a novel information-theoretic framework designed to enhance the process of machine unlearning that effectively leverages the influence of systematic patterns and biases for parameter adjustment. By proposing a variational upper bound, we recalibrate the model parameters through a dynamic prior that integrates changes in data distribution with an affordable computational cost, allowing efficient and accurate removal of outdated or unwanted data patterns and biases. Our experiments across various datasets, models, and unlearning methods demonstrate that our approach effectively removes systematic patterns and biases while maintaining the performance of models post-unlearning.