# Variational (Gradient) Estimate of the Score Function in Energy-based Latent Variable Models

arXiv: https://arxiv.org/abs/2010.08258 · HTML: https://mlsanity.com/paper/2010.08258/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2020.

## Built on by (most cited first, 8 of 8)

- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Offline Reinforcement Learning with Fisher Divergence Critic Regularization](https://mlsanity.com/paper/2103.08050/) (2021)
- [Concrete Score Matching: Generalized Score Matching for Discrete Data](https://mlsanity.com/paper/2211.00802/) (2022)
- [AI-Generated Images as Data Source: The Dawn of Synthetic Era](https://mlsanity.com/paper/2310.01830/) (2023)
- [Score Matching With Missing Data](https://mlsanity.com/paper/2506.00557/) (2025)
- [Bi-level Doubly Variational Learning for Energy-based Latent Variable Models](https://mlsanity.com/paper/2203.14702/) (2022)
- [A Comprehensive Review of Diffusion Models in Smart Agriculture: Progress, Applications, and Challenges](https://mlsanity.com/paper/2507.18376/) (2025)
- [Self-Adapting Noise-Contrastive Estimation for Energy-Based Models](https://mlsanity.com/paper/2211.02650/) (2022)

## Abstract (arXiv)

The learning and evaluation of energy-based latent variable models (EBLVMs) without any structural assumptions are highly challenging, because the true posteriors and the partition functions in such models are generally intractable. This paper presents variational estimates of the score function and its gradient with respect to the model parameters in a general EBLVM, referred to as VaES and VaGES respectively. The variational posterior is trained to minimize a certain divergence to the true model posterior and the bias in both estimates can be bounded by the divergence theoretically. With a minimal model assumption, VaES and VaGES can be applied to the kernelized Stein discrepancy (KSD) and score matching (SM)-based methods to learn EBLVMs. Besides, VaES can also be used to estimate the exact Fisher divergence between the data and general EBLVMs.