# Provable Bayesian Inference via Particle Mirror Descent

arXiv: https://arxiv.org/abs/1506.03101 · HTML: https://mlsanity.com/paper/1506.03101/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2015.

## Built on by (most cited first, 23 of 23)

- [Inference-Aware Fine-Tuning for Best-of-N Sampling in Large Language Models](https://mlsanity.com/paper/2412.15287/) (2024)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [Batch and match: black-box variational inference with a score-based divergence](https://mlsanity.com/paper/2402.14758/) (2024)
- [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023)
- [Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff](https://mlsanity.com/paper/2110.15590/) (2021)
- [Annealing Flow Generative Models Towards Sampling High-Dimensional and Multi-Modal Distributions](https://mlsanity.com/paper/2409.20547/) (2024)
- [Inferring Smooth Control: Monte Carlo Posterior Policy Iteration with Gaussian Processes](https://mlsanity.com/paper/2210.03512/) (2022)
- [Interaction-Force Transport Gradient Flows](https://mlsanity.com/paper/2405.17075/) (2024)
- [Adaptive Annealed Importance Sampling with Constant Rate Progress](https://mlsanity.com/paper/2306.15283/) (2023)
- [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024)
- [Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals](https://mlsanity.com/paper/2501.17049/) (2025)
- [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025)
- [Analysis of Kernel Mirror Prox for Measure Optimization](https://mlsanity.com/paper/2403.00147/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025)
- [DF2: Distribution-Free Decision-Focused Learning](https://mlsanity.com/paper/2308.05889/) (2023)
- [A Quadrature Rule combining Control Variates and Adaptive Importance Sampling](https://mlsanity.com/paper/2205.11890/) (2022)
- [Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization](https://mlsanity.com/paper/2512.12550/) (2025)
- [Adaptive Bayesian Online Learning via Expert Aggregation](https://mlsanity.com/paper/2607.20239/) (2026)
- [Pseudo-Likelihood Inference](https://mlsanity.com/paper/2311.16656/) (2023)
- [Approximated Orthogonal Projection Unit: Stabilizing Regression Network Training Using Natural Gradient](https://mlsanity.com/paper/2409.15393/) (2024)
- [Entropic Mirror Monte Carlo](https://mlsanity.com/paper/2602.03165/) (2026)
- [Fast rates in Bayesian online learning with approximate posteriors](https://mlsanity.com/paper/2608.25706/) (2026)

## Abstract (arXiv)

Bayesian methods are appealing in their flexibility in modeling complex data and ability in capturing uncertainty in parameters. However, when Bayes' rule does not result in tractable closed-form, most approximate inference algorithms lack either scalability or rigorous guarantees. To tackle this challenge, we propose a simple yet provable algorithm, Particle Mirror Descent (PMD), to iteratively approximate the posterior density. PMD is inspired by stochastic functional mirror descent where one descends in the density space using a small batch of data points at each iteration, and by particle filtering where one uses samples to approximate a function. We prove result of the first kind that, with m particles, PMD provides a posterior density estimator that converges in terms of KL-divergence to the true posterior in rate O(1/√m). We demonstrate competitive empirical performances of PMD compared to several approximate inference algorithms in mixture models, logistic regression, sparse Gaussian processes and latent Dirichlet allocation on large scale datasets.