Scene coordinate regression (SCR) models have proven to be powerful implicit scene representations for 3D vision, enabling visual relocalization and structure-from-motion. SCR models are trained specifically for one scene. If training images imply insufficient multi-view constraints to recover the scene geometry, SCR models degenerate. We present a probabilistic reinterpretation of training SCR models, which allows us to infuse high-level reconstruction priors. We investigate multiple such priors, ranging from simple priors over the distribution of reconstructed depth values to learned priors over plausible scene coordinate configurations. For the latter, we train a 3D point cloud diffusion model on a large corpus of indoor scans. Our priors push predicted 3D scene points towards a more plausible geometry at each training step to increase their likelihood. On three indoor datasets our priors help learning better scene representations, resulting in more coherent scene point clouds, higher registration rates and better camera poses, with a positive effect on down-stream tasks such as novel view synthesis and camera relocalization.