MCMC-Interactive Variational Inference

TL;DR

MCMC-Interactive Variational Inference (MIVI) combines variational inference and MCMC to efficiently estimate posteriors, improve uncertainty quantification, and guide the design of MCMC transitions, with theoretical guarantees of convergence.

Contribution

MIVI introduces a novel framework that jointly optimizes a variational distribution and a Markov chain, enhancing posterior approximation and transition design.

Findings

Accurately approximates complex posteriors with fewer transitions.

Facilitates the design of stochastic gradient MCMC and Gibbs sampling.

Proves the Markov chain's extrapolation property improves posterior convergence.

Abstract

Leveraging well-established MCMC strategies, we propose MCMC-interactive variational inference (MIVI) to not only estimate the posterior in a time constrained manner, but also facilitate the design of MCMC transitions. Constructing a variational distribution followed by a short Markov chain that has parameters to learn, MIVI takes advantage of the complementary properties of variational inference and MCMC to encourage mutual improvement. On one hand, with the variational distribution locating high posterior density regions, the Markov chain is optimized within the variational inference framework to efficiently target the posterior despite a small number of transitions. On the other hand, the optimized Markov chain with considerable flexibility guides the variational distribution towards the posterior and alleviates its underestimation of uncertainty. Furthermore, we prove the optimized Markov chain in MIVI admits extrapolation, which means its marginal distribution gets closer to the true posterior as the chain grows. Therefore, the Markov chain can be used separately as an efficient MCMC scheme. Experiments show that MIVI not only accurately and efficiently approximates the posteriors but also facilitates designs of stochastic gradient MCMC and Gibbs sampling transitions.