Markov Chain Monte Carlo and Variational Inference: Bridging the Gap
Tim Salimans Algoritmica Diederik P. Kingma and Max Welling University of Amsterdam
Abstract
Recent advances in stochastic gradient variational inference have made it possible to perform variational Bayesian inference with posterior approximations containing auxiliary random variables. This enables us to explore a new synthesis of variational inference and Monte Carlo methods where we incorporate one or more steps of MCMC into our variational approximation. By doing so we obtain a rich class of inference algorithms bridging the gap between variational methods and MCMC, and offering the best of both worlds: fast posterior approximation through the maximization of an explicit objective, with the option of trading off additional computation for additional accuracy. We describe the theoretical foundations that make this possible and show some promising first results.
中文速览
变分推断（variational inference）速度快但精度受限于近似族的选择，而马尔可夫链蒙特卡洛（MCMC）渐近精确却慢得难以实用，这篇论文想把两者的优点合而为一。核心思路是把MCMC的马尔可夫链转移步骤重新解读为一种带辅助变量的变分近似，从而把每一步MCMC嵌入变分下界（variational lower bound）的优化框架里，同时引入一个"逆向模型"来弥补辅助变量带来的近似损失。在此基础上，作者重点提出了"哈密顿变分推断"（Hamiltonian Variational Inference, HVI），将哈密顿蒙特卡洛（HMC）的高效探索能力与变分推断的显式目标函数优化相结合，并通过重参数化技巧对下界做无偏随机梯度估计。在双变量高斯采样和密苏里胃癌死亡率贝塔-二项模型两个实验中，HVI都比标准变分近似给出了更接近真实后验的结果，而且哈密顿动力学步骤有时还能降低梯度方差、加快收敛。这项工作的意义在于提供了一个统一框架，让研究者可以用计算量换精度——只需增加MCMC步数，就能在不牺牲优化速度优势的前提下持续逼近精确后验。
原文 arXiv:1410.6460;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1410.6460v4