# Variational MCMC

arXiv: https://arxiv.org/abs/1301.2266 · HTML: https://mlsanity.com/paper/1301.2266/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2013.

## Built on by (most cited first, 7 of 7)

- [Structured Stochastic Gradient MCMC](https://mlsanity.com/paper/2107.09028/) (2021)
- [Interpolating between sampling and variational inference with infinite stochastic mixtures](https://mlsanity.com/paper/2110.09618/) (2021)
- [A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors](https://mlsanity.com/paper/2111.13329/) (2021)
- [Semi-Empirical Objective Functions for MCMC Proposal Optimization](https://mlsanity.com/paper/2106.02104/) (2021)
- [Fitting Large Nonlinear Mixed Effects Models Using Variational Expectation Maximization](https://mlsanity.com/paper/2604.26160/) (2026)
- [STANLEY: Stochastic Gradient Anisotropic Langevin Dynamics for Learning Energy-Based Models](https://mlsanity.com/paper/2310.12667/) (2023)
- [Neural-Inspired Posterior Approximation (NIPA)](https://mlsanity.com/paper/2601.22539/) (2026)

## Abstract (arXiv)

We propose a new class of learning algorithms that combines variational approximation and Markov chain Monte Carlo (MCMC) simulation. Naive algorithms that use the variational approximation as proposal distribution can perform poorly because this approximation tends to underestimate the true variance and other features of the data. We solve this problem by introducing more sophisticated MCMC algorithms. One of these algorithms is a mixture of two MCMC kernels: a random walk Metropolis kernel and a blockMetropolis-Hastings (MH) kernel with a variational approximation as proposaldistribution. The MH kernel allows one to locate regions of high probability efficiently. The Metropolis kernel allows us to explore the vicinity of these regions. This algorithm outperforms variationalapproximations because it yields slightly better estimates of the mean and considerably better estimates of higher moments, such as covariances. It also outperforms standard MCMC algorithms because it locates theregions of high probability quickly, thus speeding up convergence. We demonstrate this algorithm on the problem of Bayesian parameter estimation for logistic (sigmoid) belief networks.