Variational Inference

What is Variational Inference? Variational Inference Explained

Variational inference is a technique used in Bayesian statistics and machine learning to approximate complex posterior distributions when exact inference is intractable. It provides a way to perform efficient and scalable inference by formulating it as an optimization problem.

In Bayesian inference, the goal is to compute the posterior distribution of model parameters given the observed data. However, in many cases, the exact computation of the posterior distribution is analytically or computationally challenging, especially when dealing with high-dimensional or complex models.

Variational inference addresses this challenge by introducing an approximating distribution, often referred to as a variational distribution or a recognition model. This approximating distribution is typically chosen from a tractable family, such as a Gaussian distribution, and is parameterized by variational parameters.

The main idea of variational inference is to find the best approximation to the true posterior by minimizing the Kullback-Leibler (KL) divergence between the approximating distribution and the true posterior. The KL divergence measures the dissimilarity between two probability distributions. By minimizing this divergence, variational inference seeks an approximating distribution that is as close as possible to the true posterior.

To find the optimal variational parameters, variational inference formulates an optimization problem known as the variational lower bound or evidence lower bound (ELBO). The ELBO is derived by decomposing the logarithm of the posterior distribution and applying Jensen’s inequality. Maximizing the ELBO with respect to the variational parameters is equivalent to minimizing the KL divergence.

The optimization process in variational inference typically involves iterative updates of the variational parameters using techniques like gradient descent or stochastic optimization. The updates aim to maximize the ELBO, which leads to improving the approximation of the posterior distribution.

Once the optimization converges, the approximating distribution can be used to approximate the posterior distribution for various purposes, such as computing posterior summaries, making predictions, or performing inference on new data.

Variational inference offers several advantages, including scalability to large datasets and complex models, flexibility in choosing the form of the approximating distribution, and the ability to handle missing data or latent variables. However, it introduces an approximation error due to the use of a simpler approximating distribution.

Overall, variational inference provides a powerful framework for approximating complex posterior distributions and performing efficient Bayesian inference. It has found applications in various domains, including machine learning, Bayesian statistics, and probabilistic modeling.