Variational inference is a method from machine learning used to approximate probability densities. It is particularly useful in Bayesian statistics for approximating posterior densities, often being faster than Markov chain Monte Carlo (MCMC) sampling. The core idea is to find a family of densities that minimizes the Kullback-Leibler (KL) divergence to the target posterior. Mean-field variational inference, a special case, assumes independence among latent variables and uses optimization to find the best approximation. Variational inference can be extended to handle large data using stochastic optimization. It is more efficient than MCMC but less rigorously studied. The paper reviews variational inference, discusses its applications, and highlights open problems. It also provides a detailed example of Bayesian mixture of Gaussians and discusses the relationship between variational inference and MCMC. The paper emphasizes that variational inference is a valuable tool for approximate Bayesian inference, especially for large datasets and complex models.Variational inference is a method from machine learning used to approximate probability densities. It is particularly useful in Bayesian statistics for approximating posterior densities, often being faster than Markov chain Monte Carlo (MCMC) sampling. The core idea is to find a family of densities that minimizes the Kullback-Leibler (KL) divergence to the target posterior. Mean-field variational inference, a special case, assumes independence among latent variables and uses optimization to find the best approximation. Variational inference can be extended to handle large data using stochastic optimization. It is more efficient than MCMC but less rigorously studied. The paper reviews variational inference, discusses its applications, and highlights open problems. It also provides a detailed example of Bayesian mixture of Gaussians and discusses the relationship between variational inference and MCMC. The paper emphasizes that variational inference is a valuable tool for approximate Bayesian inference, especially for large datasets and complex models.