This paper reviews variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI is widely used in Bayesian statistics to approximate posterior densities, offering a faster alternative to Markov chain Monte Carlo (MCMC) sampling. The authors discuss the basic ideas behind mean-field VI, the special case of VI for exponential family models, and provide a full example using a Bayesian mixture of Gaussians. They also introduce stochastic optimization to scale VI to large datasets. The paper highlights the importance of understanding the statistical properties of VI and outlines open research directions. VI is powerful but not fully understood, and the authors aim to catalyze statistical research on this class of algorithms.This paper reviews variational inference (VI), a method from machine learning that approximates probability densities through optimization. VI is widely used in Bayesian statistics to approximate posterior densities, offering a faster alternative to Markov chain Monte Carlo (MCMC) sampling. The authors discuss the basic ideas behind mean-field VI, the special case of VI for exponential family models, and provide a full example using a Bayesian mixture of Gaussians. They also introduce stochastic optimization to scale VI to large datasets. The paper highlights the importance of understanding the statistical properties of VI and outlines open research directions. VI is powerful but not fully understood, and the authors aim to catalyze statistical research on this class of algorithms.