# The Implicit Metropolis-Hastings Algorithm

## TL;DR

The paper introduces the implicit Metropolis-Hastings algorithm, which uses a learned discriminator to generate samples approximating a target distribution, with theoretical guarantees on its stationary distribution and validation on image datasets.

## Contribution

It generalizes GAN filtering ideas by developing a new sampling algorithm with theoretical analysis and empirical validation on real datasets.

## Key findings

- Discriminator loss bounds the total variation distance to the target distribution.
- The algorithm effectively generates samples close to the target distribution.
- Validation on CIFAR-10 and CelebA demonstrates practical applicability.

## Abstract

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.