Precise estimation of uncertainty in predictions for AI systems is a critical factor in ensuring trust and safety. Deep neural networks trained with a conventional method are prone to over-confident predictions. In contrast to Bayesian neural networks that learn approximate distributions on weights to infer prediction confidence, we propose a novel method, Information Aware Dirichlet networks, that learn an explicit Dirichlet prior distribution on predictive distributions by minimizing a bound on the expected max norm of the prediction error and penalizing information associated with incorrect outcomes. Properties of the new cost function are derived to indicate how improved uncertainty estimation is achieved. Experiments using real datasets show that our technique outperforms, by a large margin, state-of-the-art neural networks for estimating within-distribution and out-of-distribution uncertainty, and detecting adversarial examples.