Variational Inference: A Review for Statisticians

TL;DR

This paper reviews variational inference, a machine learning technique for approximating complex probability densities, highlighting its methods, applications, and open research challenges to motivate further statistical research.

Contribution

It provides a comprehensive overview of variational inference, including mean-field methods, exponential family models, stochastic optimization, and discusses current research and open problems.

Findings

VI is faster than traditional methods like MCMC.

Mean-field VI simplifies complex models effectively.

Stochastic VI scales to large datasets.

Abstract

One of the core problems of modern statistics is to approximate difficult-to-compute probability densities. This problem is especially important in Bayesian statistics, which frames all inference about unknown quantities as a calculation involving the posterior density. In this paper, we review variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI has been used in many applications and tends to be faster than classical methods, such as Markov chain Monte Carlo sampling. The idea behind VI is to first posit a family of densities and then to find the member of that family which is close to the target. Closeness is measured by Kullback-Leibler divergence. We review the ideas behind mean-field variational inference, discuss the special case of VI applied to exponential family models, present a full example with a Bayesian mixture of Gaussians, and derive a variant that uses stochastic optimization to scale up to massive data. We discuss modern research in VI and highlight important open problems. VI is powerful, but it is not yet well understood. Our hope in writing this paper is to catalyze statistical research on this class of algorithms.