# Interpolating between sampling and variational inference with infinite stochastic mixtures

arXiv: https://arxiv.org/abs/2110.09618 · HTML: https://mlsanity.com/paper/2110.09618/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 55 references that are in the map.
- First posted on arXiv 18 Oct 2021.

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [On Variational Bounds of Mutual Information](https://mlsanity.com/paper/1905.06922/) (2019)
- [Advances in Variational Inference](https://mlsanity.com/paper/1711.05597/) (2017)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [Hierarchical Variational Models](https://mlsanity.com/paper/1511.02386/) (2015)
- [Semi-Implicit Variational Inference](https://mlsanity.com/paper/1805.11183/) (2018)
- [Variational Boosting: Iteratively Refining Posterior Approximations](https://mlsanity.com/paper/1611.06585/) (2016)
- [Estimating Mixture Entropy with Pairwise Distances](https://mlsanity.com/paper/1706.02419/) (2017)
- [Pathfinder: Parallel quasi-Newton variational inference](https://mlsanity.com/paper/2108.03782/) (2021)
- [Patterns of Scalable Bayesian Inference](https://mlsanity.com/paper/1602.05221/) (2016)
- [Austerity in MCMC Land: Cutting the Metropolis-Hastings Budget](https://mlsanity.com/paper/1304.5299/) (2013)
- [Variational Bayesian Monte Carlo with Noisy Likelihoods](https://mlsanity.com/paper/2006.08655/) (2020)
- [Boosting Variational Inference](https://mlsanity.com/paper/1611.05559/) (2016)
- [Variational Bayesian Monte Carlo](https://mlsanity.com/paper/1810.05558/) (2018)
- [Variational Inference with Hamiltonian Monte Carlo](https://mlsanity.com/paper/1609.08203/) (2016)
- [Variational MCMC](https://mlsanity.com/paper/1301.2266/) (2013)

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

- [High Dimensional Causal Inference with Variational Backdoor Adjustment](https://mlsanity.com/paper/2310.06100/) (2023)
- [Inferring Inference](https://mlsanity.com/paper/2310.03186/) (2023)
- [How does the brain compute with probabilities?](https://mlsanity.com/paper/2409.02709/) (2024)

## Abstract (arXiv)

Sampling and Variational Inference (VI) are two large families of methods for approximate inference that have complementary strengths. Sampling methods excel at approximating arbitrary probability distributions, but can be inefficient. VI methods are efficient, but may misrepresent the true distribution. Here, we develop a general framework where approximations are stochastic mixtures of simple component distributions. Both sampling and VI can be seen as special cases: in sampling, each mixture component is a delta-function and is chosen stochastically, while in standard VI a single component is chosen to minimize divergence. We derive a practical method that interpolates between sampling and VI by solving an optimization problem over a mixing distribution. Intermediate inference methods then arise by varying a single parameter. Our method provably improves on sampling (reducing variance) and on VI (reducing bias+variance despite increasing variance). We demonstrate our method's bias/variance trade-off in practice on reference problems, and we compare outcomes to commonly used sampling and VI methods. This work takes a step towards a highly flexible yet simple family of inference methods that combines the complementary strengths of sampling and VI.