Bayesian inverse reinforcement learning (IRL) methods are ideal for safe imitation learning, as they allow a learning agent to reason about reward uncertainty and the safety of a learned policy. However, Bayesian IRL is computationally intractable for high-dimensional problems because each sample from the posterior requires solving an entire Markov Decision Process (MDP). While there exist non-Bayesian deep IRL methods, these methods typically infer point estimates of reward functions, precluding rigorous safety and uncertainty analysis. We propose Bayesian Reward Extrapolation (B-REX), a highly efficient, preference-based Bayesian reward learning algorithm that scales to high-dimensional, visual control tasks. Our approach uses successor feature representations and preferences over demonstrations to efficiently generate samples from the posterior distribution over the demonstrator's reward function without requiring an MDP solver. Using samples from the posterior, we demonstrate how to calculate high-confidence bounds on policy performance in the imitation learning setting, in which the ground-truth reward function is unknown. We evaluate our proposed approach on the task of learning to play Atari games via imitation learning from pixel inputs, with no access to the game score. We demonstrate that B-REX learns imitation policies that are competitive with a state-of-the-art deep imitation learning method that only learns a point estimate of the reward function. Furthermore, we demonstrate that samples from the posterior generated via B-REX can be used to compute high-confidence performance bounds for a variety of evaluation policies. We show that high-confidence performance bounds are useful for accurately ranking different evaluation policies when the reward function is unknown. We also demonstrate that high-confidence performance bounds may be useful for detecting reward hacking.