Energy-Tweedie: Score meets Score, Energy meets Energy

Mon, 29 Dec 2025 00:00:00 +0000

Classical Tweedie’s formula links Gaussian corruption, squared-error denoising, the posterior mean, and the score of the noisy data. The Energy–Tweedie identity generalizes this correspondence from a mean-based relation to a distributional one. For generalized Gaussian corruption, it connects the noisy-data score to the path derivative of a matched energy score—a scoring rule evaluated at the full denoising posterior. In the Gaussian case, this reduces exactly to classical Tweedie’s formula.

The identity has three main consequences:

Samples from a learned denoising posterior can be converted into a score estimate.
The identity can be used to recover the parameters of the noise distribution.
This provides a score-based perspective on diffusion approaches based on scoring rules; existing score-based samplers can thus be used to generate from such models.

We validate these consequences in controlled experiments covering score recovery, noise-parameter estimation, and diffusion-style sampling.

Scoring Rules | Andrej Leban

Energy-Tweedie: Score meets Score, Energy meets Energy