Title: Mixture of Basis for Interpretable Continual Learning with Distribution Shifts. Abstract: Continual learning in environments with shifting data distributions is a challenging problem with several real-world applications. In this paper we consider settings in which the data distribution (i.e., task) shifts abruptly and the timing of these shifts are not known. Furthermore, we consider a $\textit{semi-supervised task-agnostic}$ setting in which the learning algorithm has access to both task-segmented and unsegmented data for offline training. We propose a novel approach called $\textit{Mixture of Basis}$ models $\textit{(MoB)}$ for addressing this problem setting. The core idea is to learn a small set of $\textit{basis models}$ and to construct a dynamic, task-dependent mixture of the models to predict for the current task. We also propose a new methodology to detect observations that are out-of-distribution with respect to the existing basis models and to instantiate new models as needed. We develop novel problem domains for regression tasks, evaluate MoB and other continual learning algorithms on these, and show that MoB attains better prediction error in nearly every case while using fewer models than other multiple-model approaches. We analyze latent task representations learned by MoB alongside the tasks themselves, using both qualitative and quantitative measures, to show that the learned latent task representations can be interpretably linked to the structure of the task space.