Variational Bayes

What is Variational Bayes? Variational Bayes Explained

Variational Bayes (VB) is a technique used in Bayesian inference to approximate the posterior distribution of model parameters when exact inference is intractable. It provides a computationally efficient way to perform Bayesian inference by formulating it as an optimization problem.

In Bayesian inference, the goal is to compute the posterior distribution of the model parameters given the observed data. However, in many complex models, the posterior distribution cannot be computed analytically due to the high-dimensional and nonlinear nature of the problem. Variational Bayes offers a practical approximation to the posterior distribution by introducing a simpler distribution that is easier to work with.

The core idea of Variational Bayes is to cast the posterior inference problem as an optimization problem. Instead of directly computing the true posterior distribution, VB introduces a family of approximating distributions, typically parameterized by a set of variational parameters. The goal is to find the best approximation within this family that is closest to the true posterior.

The optimization problem is formulated by minimizing the Kullback-Leibler (KL) divergence between the approximating distribution and the true posterior. The KL divergence measures the dissimilarity between two distributions. By minimizing this divergence, Variational Bayes finds the best approximation that balances closeness to the true posterior and computational simplicity.

The optimization process in Variational Bayes involves iteratively updating the variational parameters to minimize the KL divergence. This can be done using various optimization algorithms, such as gradient descent or coordinate descent. The updates are typically performed by maximizing a lower bound on the log-likelihood of the observed data, known as the evidence lower bound (ELBO).

Once the variational optimization converges, the approximating distribution can be used to approximate the posterior distribution. This allows for efficient computation of posterior summaries, such as mean, variance, or quantiles, as well as performing various downstream analyses and predictions.

Variational Bayes has become a popular technique in machine learning and Bayesian statistics, as it provides a scalable and computationally efficient approach to Bayesian inference in complex models. It allows practitioners to perform approximate inference even when exact solutions are not feasible, enabling the application of Bayesian methods to a wide range of problems.