Energy-Tweedie: Score meets Score, Energy meets Energy

Denoising and score estimation are classically linked through Tweedie’s formula, which relates the posterior mean under Gaussian corruption to the Stein score of the noisy marginal. In this work, we extend this perspective beyond Gaussian noise to a broad class of elliptical, energy-based noise distributions, with particular emphasis on generalized Gaussian corruptions. We derive the Energy–Tweedie identity: when the denoising posterior is viewed through the lens of proper scoring rules, the path derivative of a matched, possibly non-Euclidean energy score recovers the Stein score of the noisy marginal. Thus, the familiar correspondence between Gaussian noise, posterior means, squared loss, and Tweedie’s formula is lifted to a distributional correspondence between generalized Gaussian noise, full posterior laws, Mahalanobis energy scores, and the Energy–Tweedie identity. Among its consequences, this identity gives a posterior-sample-based route to score estimation, yields a principled criterion for noise-parameter calibration, and supplies a score-based perspective on recent diffusion-style generative methods trained with scoring rules.