# Entropic GANs meet VAEs: A Statistical Approach to Compute Sample Likelihoods in GANs

arXiv: https://arxiv.org/abs/1810.04147 · HTML: https://mlsanity.com/paper/1810.04147/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2018.

## Built on by (most cited first, 4 of 4)

- [Extracting Training Data from Diffusion Models](https://mlsanity.com/paper/2301.13188/) (2023)
- [Understanding Entropic Regularization in GANs](https://mlsanity.com/paper/2111.01387/) (2021)
- [Symmetric Wasserstein Autoencoders](https://mlsanity.com/paper/2106.13024/) (2021)
- [Bridging GANs and Bayesian Neural Networks via Partial Stochasticity](https://mlsanity.com/paper/2507.00651/) (2025)

## Abstract (arXiv)

Building on the success of deep learning, two modern approaches to learn a probability model from the data are Generative Adversarial Networks (GANs) and Variational AutoEncoders (VAEs). VAEs consider an explicit probability model for the data and compute a generative distribution by maximizing a variational lower-bound on the log-likelihood function. GANs, however, compute a generative model by minimizing a distance between observed and generated probability distributions without considering an explicit model for the observed data. The lack of having explicit probability models in GANs prohibits computation of sample likelihoods in their frameworks and limits their use in statistical inference problems. In this work, we resolve this issue by constructing an explicit probability model that can be used to compute sample likelihood statistics in GANs. In particular, we prove that under this probability model, a family of Wasserstein GANs with an entropy regularization can be viewed as a generative model that maximizes a variational lower-bound on average sample log likelihoods, an approach that VAEs are based on. This result makes a principled connection between two modern generative models, namely GANs and VAEs. In addition to the aforementioned theoretical results, we compute likelihood statistics for GANs trained on Gaussian, MNIST, SVHN, CIFAR-10 and LSUN datasets. Our numerical results validate the proposed theory.