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One notable weakness of current machine learning algorithms is the poor ability of models to solve new problems without forgetting previously acquired knowledge. The Continual Learning paradigm has emerged as a protocol to systematically investigate settings where the model sequentially observes samples generated by a series of tasks. In this work, we take a task-agnostic view of continual learning and develop a hierarchical information-theoretic optimality principle that facilitates a trade-off between learning and forgetting. We derive this principle from a Bayesian perspective and show its connections to previous approaches to continual learning. Based on this principle, we propose a neural network layer, called the Mixture-of-Variational-Experts layer, that alleviates forgetting by creating a set of information processing paths through the network which is governed by a gating policy. Equipped with a diverse and specialized set of parameters, each path can be regarded as a distinct sub-network that learns to solve tasks. To improve expert allocation, we introduce diversity objectives, which we evaluate in additional ablation studies. Importantly, our approach can operate in a task-agnostic way, i.e., it does not require task-specific knowledge, as is the case with many existing continual learning algorithms. Due to the general formulation based on generic utility functions, we can apply this optimality principle to a large variety of learning problems, including supervised learning, reinforcement learning, and generative modeling. We demonstrate the competitive performance of our method on continual reinforcement learning and variants of the MNIST, CIFAR-10, and CIFAR-100 datasets.