We introduce a new, rigorously-formulated Bayesian meta-learning algorithm that learns a probability distribution of model parameter prior for few-shot learning. The proposed algorithm employs a gradient-based variational inference to infer the posterior of model parameters to a new task. Our algorithm can be applied to any model architecture and can be implemented in various machine learning paradigms, including regression and classification. We show that the models trained with our proposed meta-learning algorithm are well calibrated and accurate, with state-of-the-art calibration and classification results on two few-shot classification benchmarks (Omniglot and Mini-ImageNet), and competitive results in a multi-modal task-distribution regression.