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Energy-Tweedie: Score meets Score, Energy meets Energy

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Posterior sampling with diffusion guidance · 0 citations (OpenAlex)

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Derives an Energy-Tweedie identity linking the Stein score of a noisy marginal under Gibbs (energy-based, e.g. generalized Gaussian) noise to the path derivative of a kernel scoring rule evaluated at the denoising posterior, and applies it to score estimation, noise-parameter M-estimation, and diffusion-style sampling along noise-parameter paths.

Problem

Tweedie's formula ties posterior means and scores only under Gaussian noise; for non-Gaussian (Gibbs/energy-based) additive noise the posterior mean is no longer the natural object exposed by the noisy marginal score.

Approach

For additive Gibbs noise with potential E, the Stein score of the noisy marginal equals the negative path derivative of the kernel score induced by rho(x,y)=E(y-x) evaluated at the full denoising posterior, giving one Tweedie-style identity per noise potential; propriety depends only on E (CND / strong negative type).

Key results

• Score NMSE 0.029-0.077 and cosine similarity 0.951-0.982 across five one-parameter sweeps of the generalized Gaussian posterior-based score on Eight-Gaussians.
• Fully learned generalized fit recovers (beta, lambda, u, v) = (1.44, 1.79, 0.25, 0.35) vs true (1.4, 1.8, 0.25, 0.35) with held-out NMSE 0.0146, lower than all misspecified fits (0.0342 to 0.2081).
• On MNIST, a single posterior model drives annealed Langevin sampling along five noise-parameter paths producing recognizable, class-diverse digits; generalized/non-diagonal path gets best feature energy distance.

Surprising findings

• The Student-t case gives a valid score identity even though the noise mean is undefined, with a strictly proper scoring rule.
• Forced isotropic generalized fits drag beta down to 1.12 to mimic missing anisotropy, showing the fitted family rather than component quality limits accuracy.

Limitations

• Identity is exact only under the assumed additive Gibbs noise model and correctly specified posterior.
• Experiments restricted to the generalized Gaussian family on low-dimensional synthetic data and MNIST; no large-scale image generation benchmarks.
• Noise-parameter estimation consistency assumes accurate auxiliary score estimator and posterior.

Experimental setup

Datasets: Eight-Gaussians synthetic 2D MNIST

Benchmarks: Eight-Gaussians MNIST

Metrics: score NMSE cosine similarity posterior-mean MSE parameter estimation error feature energy distance

Baselines: Gaussian Tweedie / denoising score matching forced Gaussian anisotropic fit forced isotropic generalized fit forced isotropic Gaussian fit

Builds on: Tweedie's formula energy score Engression Fisher identity annealed Langevin dynamics

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