# Efficient Bayesian Sampling Using Normalizing Flows to Assist Markov Chain Monte Carlo Methods

arXiv: https://arxiv.org/abs/2107.08001 · HTML: https://mlsanity.com/paper/2107.08001/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 16 Jul 2021.

## Builds on (most cited first, 14 of 14)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Neural Importance Sampling](https://mlsanity.com/paper/1808.03856/) (2018)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Solving Statistical Mechanics Using Variational Autoregressive Networks](https://mlsanity.com/paper/1809.10606/) (2018)
- [Generalizing Hamiltonian Monte Carlo with Neural Networks](https://mlsanity.com/paper/1711.09268/) (2017)
- [A-NICE-MC: Adversarial Training for MCMC](https://mlsanity.com/paper/1706.07561/) (2017)
- [Scalable Nonparametric Sampling from Multimodal Posteriors with the Posterior Bootstrap](https://mlsanity.com/paper/1902.03175/) (2019)
- [Learning Model Reparametrizations: Implicit Variational Inference by Fitting MCMC distributions](https://mlsanity.com/paper/1708.01529/) (2017)
- [Self-Supervised Learning of Generative Spin-Glasses with Normalizing Flows](https://mlsanity.com/paper/2001.00585/) (2020)

## Built on by (most cited first, 16 of 16)

- [Sequential Controlled Langevin Diffusions](https://mlsanity.com/paper/2412.07081/) (2024)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster training](https://mlsanity.com/paper/2501.06148/) (2025)
- [Amortized Sampling with Transferable Normalizing Flows](https://mlsanity.com/paper/2508.18175/) (2025)
- [Wavelet Conditional Renormalization Group](https://mlsanity.com/paper/2207.04941/) (2022)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Accelerating Markov Chain Monte Carlo sampling with diffusion models](https://mlsanity.com/paper/2309.01454/) (2023)
- [Markovian Flow Matching: Accelerating MCMC with Continuous Normalizing Flows](https://mlsanity.com/paper/2405.14392/) (2024)
- [Mitigating mode collapse in normalizing flows by annealing with an adaptive schedule: Application to parameter estimation](https://mlsanity.com/paper/2505.03652/) (2025)
- [Reconstructing the Universe with Variational self-Boosted Sampling](https://mlsanity.com/paper/2206.15433/) (2022)
- [Enhanced Sampling with Machine Learning: A Review](https://mlsanity.com/paper/2306.09111/) (2023)
- [Leveraging generative models to assist Monte Carlo sampling](https://mlsanity.com/paper/2608.07648/) (2026)
- [Are Normalizing Flows the Key to Unlocking the Exponential Mechanism?](https://mlsanity.com/paper/2311.09200/) (2023)
- [Multi-Armed Sampling Problem and the End of Exploration](https://mlsanity.com/paper/2507.10797/) (2025)
- [Reducing normalizing flow complexity for MCMC preconditioning](https://mlsanity.com/paper/2511.02345/) (2025)
- [A Novel Framework Using Variational Inference with Normalizing Flows to Train Transport Reversible Jump Proposals](https://mlsanity.com/paper/2512.12742/) (2025)

## Abstract (arXiv)

Normalizing flows can generate complex target distributions and thus show promise in many applications in Bayesian statistics as an alternative or complement to MCMC for sampling posteriors. Since no data set from the target posterior distribution is available beforehand, the flow is typically trained using the reverse Kullback-Leibler (KL) divergence that only requires samples from a base distribution. This strategy may perform poorly when the posterior is complicated and hard to sample with an untrained normalizing flow. Here we explore a distinct training strategy, using the direct KL divergence as loss, in which samples from the posterior are generated by (i) assisting a local MCMC algorithm on the posterior with a normalizing flow to accelerate its mixing rate and (ii) using the data generated this way to train the flow. The method only requires a limited amount of textita~priori input about the posterior, and can be used to estimate the evidence required for model validation, as we illustrate on examples.