The classical Tweedie’s formula connects the score under Gaussian noise to the posterior mean. We generalize this result to the Energy-Tweedie identity: for “energy-based” (Gibbs) noising distributions, the score is connected to a path-derivative of a kernel scoring rule induced by the noise distribution itself and evaluated at the denoising posterior. Among other things, this opens new routes to score and noise-parameter estimation, and provides the score-based perspective on diffusion approaches based on scoring rules - with the path through the (multidimensional) noise-parameter space a free design choice at sampling time.
