# A unifying view of contrastive learning, importance sampling, and bridge sampling for energy-based models

arXiv: https://arxiv.org/abs/2604.08116 · HTML: https://mlsanity.com/paper/2604.08116/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 40 references that are in the map.
- First posted on arXiv 9 Apr 2026.

## Builds on (most cited first, 9 of 9)

- [Keystroke Dynamics as Part of Lifelogging](https://mlsanity.com/paper/2010.10144/) (2020)
- [Contrastive Clustering](https://mlsanity.com/paper/2009.09687/) (2020)
- [Introduction to Latent Variable Energy-Based Models: A Path Towards Autonomous Machine Intelligence](https://mlsanity.com/paper/2306.02572/) (2023)
- [Marginal likelihood computation for model selection and hypothesis testing: an extensive review](https://mlsanity.com/paper/2005.08334/) (2020)
- [The Optimal Noise in Noise-Contrastive Learning Is Not What You Think](https://mlsanity.com/paper/2203.01110/) (2022)
- [A survey of Monte Carlo methods for noisy and costly densities with application to reinforcement learning and ABC](https://mlsanity.com/paper/2108.00490/) (2021)
- [Recursive Pathways to Marginal Likelihood Estimation with Prior-Sensitivity Analysis](https://mlsanity.com/paper/1301.6450/) (2013)
- [Statistical applications of contrastive learning](https://mlsanity.com/paper/2204.13999/) (2022)
- [Optimality in importance sampling: a gentle survey](https://mlsanity.com/paper/2502.07396/) (2025)

## Abstract (arXiv)

In the last decades, energy-based models (EBMs) have become an important class of probabilistic models in which a component of the likelihood is intractable and therefore cannot be evaluated explicitly. Consequently, parameter estimation in EBMs is challenging for conventional inference methods. In this work, we provide a unified framework that connects noise contrastive estimation (NCE), reverse logistic regression (RLR), multiple importance sampling (MIS), and bridge sampling within the context of EBMs. We further show that these methods are equivalent under specific conditions. This unified perspective clarifies relationships among existing methods and enables the development of new estimators, with the potential to improve statistical and computational efficiency. Furthermore, this study helps elucidate the success of NCE in terms of its flexibility and robustness, while also identifying scenarios in which its performance can be further improved. Hence, rather than being a purely descriptive review, this work offers a unifying perspective and additional methodological contributions. The MATLAB code used in the numerical experiments is also made freely available to support the reproducibility of the results.