Before deploying autonomous systems in safety-critical applications, we must be able to understand and verify the safety of these systems. For cases where the risk or cost of real-world testing is prohibitive, we propose a simulation-based framework for 1) predicting ways in which an autonomous system is likely to fail and 2) automatically adjusting the system's design and control policy to preemptively mitigate those failures. Existing tools for failure prediction struggle to search over high-dimensional environmental parameters, cannot efficiently handle end-to-end testing for systems with vision in the loop, and provide little guidance on how to mitigate failures once they are discovered. We approach this problem through the lens of approximate Bayesian inference, using differentiable simulation and rendering for efficient failure case prediction and repair (and providing a gradient-free version of our algorithm for cases where a differentiable simulator is not available). We include a theoretical and empirical evaluation of the tradeoffs between gradient-based and gradient-free methods, applying our approach to a range of robotics and control problems, including optimizing search patterns for robot swarms, UAV formation control, and robust network control. Compared to optimization-based falsification methods, our method predicts a more diverse, representative set of failure modes, and we find that our use of differentiable simulation yields solutions that have up to 10x lower cost and requires up to 2x fewer iterations to converge relative to gradient-free techniques. In hardware experiments, we find that repairing control policies using our method leads to a 5x robustness improvement.