Cramer-Rao Inequality Generalizes the Equilibrium Energy Fluctuation-Response Relation to Nonequilibrium Steady States
arXiv:2608.23455
2026
Sampling
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper derives a distribution-independent Cramér–Rao inequality for nonequilibrium steady states that links the response of mean energy to inverse temperature, energy fluctuations, and Fisher information. For a neural stochastic model, inverse temperature can be implemented as a sampling temperature, diffusion noise parameter, or controllable stochasticity variable, while the energy can be a loss, score, or latent energy. The most actionable transfer is a Fisher-calibrated temperature controller that limits temperature changes according to a measurable sensitivity budget. A second transfer is a regularizer that trains beta-conditioned neural distributions to avoid excessive response to temperature perturbations.
Ideas from this paper
✗ Failed on benchmark
2026
Use the generalized Cramér–Rao relation to adapt the inverse-temperature or noise schedule of an energy-based sampler, diffusion sampler, or stochastic optimizer. The controller limits each temperature change according to the measured energy variance and Fisher information, preventing uncontrolled changes in the sampled energy distribution while allowing larger steps in insensitive regions.
Useful8/10
Difficulty4/10
Novelty7/10
Unverified
2026
Add a temperature-response constraint to stochastic neural predictors so that changes in inverse temperature cannot produce disproportionately large changes in expected loss or energy. This converts the nonequilibrium fluctuation-response inequality into a measurable robustness monitor and a regularizer for beta-conditioned stochastic representations.
Useful6/10
Difficulty5/10
Novelty6/10