Deep learning is increasingly moving towards a transfer learning paradigm whereby large ``foundation models'' are fine-tuned on downstream tasks, starting from an initialization learned on the source task. But an initialization contains relatively little information about the source task. %, and would not affect the final solution at all if we do a good job of optimization. Instead, we show that we can learn highly informative posteriors from the source task, which serves as the basis for priors that modify the whole loss surface on the downstream task. This simple modular approach enables significant performance gains and more data-efficient learning on various downstream classification and segmentation tasks, serving as a drop-in replacement for standard pre-training strategies.