Variational Inference

Variational inference turns Bayesian posterior estimation into an optimization problem, fitting a tractable distribution to approximate a typically intractable target. Common approaches include mean-field and structured coordinate ascent, stochastic-gradient ELBO optimization, amortized inference with neural networks and normalizing flows, and particle methods such as Stein variational gradient descent. These methods are used in settings from Gaussian processes and time-series models to deep learning and Bayesian inverse problems, where they can make uncertainty estimates practical at scale. A central tension is that faster, simpler approximations can miss posterior dependencies or modes and miscalibrate uncertainty, motivating more expressive families alongside stronger convergence and calibration guarantees.