Maximum Likelihood as a Path Integral of Score Matching along Langevin Diffusion

We study the maximum-likelihood gradient of energy-based models with continuous observable variables. Writing the negative-phase sample as the endpoint of a Langevin diffusion started at the data point, and applying Itô's lemma, we show that the log-likelihood gradient is equivalent to a path integral of score matching along the diffusion. Three consequences follow. First, the integrand vanishes at the equilibrium of the diffusion Markov chain, providing a closed-form target for mixing diagnostics. Second, for free energies that consist of a Gaussian term plus a locally affine score of the interactions, the integrand reduces to the $ϵ$-prediction loss of denoising diffusion with likelihood weighting. Third, energy-based models can be trained by scheduled noise injection. All three claims are verified experimentally, furthermore, we show that an energy-based model can be trained on CIFAR-10 at roughly one tenth of the per-epoch cost of persistent contrastive divergence.