# The Implicit Metropolis-Hastings Algorithm

arXiv: https://arxiv.org/abs/1906.03644 · HTML: https://mlsanity.com/paper/1906.03644/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2019.

## Built on by (most cited first, 3 of 3)

- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [A User-Guided Bayesian Framework for Ensemble Feature Selection in Life Science Applications (UBayFS)](https://mlsanity.com/paper/2104.14787/) (2021)
- [Ai-Sampler: Adversarial Learning of Markov kernels with involutive maps](https://mlsanity.com/paper/2406.02490/) (2024)

## Abstract (arXiv)

Recent works propose using the discriminator of a GAN to filter out unrealistic samples of the generator. We generalize these ideas by introducing the implicit Metropolis-Hastings algorithm. For any implicit probabilistic model and a target distribution represented by a set of samples, implicit Metropolis-Hastings operates by learning a discriminator to estimate the density-ratio and then generating a chain of samples. Since the approximation of density ratio introduces an error on every step of the chain, it is crucial to analyze the stationary distribution of such chain. For that purpose, we present a theoretical result stating that the discriminator loss upper bounds the total variation distance between the target distribution and the stationary distribution. Finally, we validate the proposed algorithm both for independent and Markov proposals on CIFAR-10 and CelebA datasets.