Variational inference

What it is

Variational inference (VI) is a method for approximating a posterior probability distribution, which is the distribution over unknown variables after data are observed. Instead of trying to calculate the posterior exactly, VI picks a family of candidate distributions, often called a variational family, and searches for the member that best matches the target distribution. This is useful when the exact posterior is hard to compute, especially in probabilistic generative models, which are models that describe how hidden variables and observations are jointly generated.

[1] Bayesian Brains and the Rényi Divergence [2] Bayesian inference in generative models [3] A primer on variational inference for physics-informed deep ...

How it works

The main idea is to turn inference into optimization. A loss or objective is defined so that improving it brings the approximate posterior closer to the true posterior. Commonly, the Kullback–Leibler (KL) divergence is minimized, or an equivalent objective such as the evidence lower bound (ELBO) is optimized. The approximate distribution is then adjusted using methods such as calculus of variations or gradient descent until it fits the posterior as well as possible within the chosen family.

[1] Bayesian Brains and the Rényi Divergence [2] Bayesian inference in generative models [3] A primer on variational inference for physics-informed deep ...

Why people use it

VI is valued because it can be computationally efficient and scalable. It can avoid expensive sampling methods such as Markov chain Monte Carlo (MCMC), which may be hard to run for complex models. In some settings, VI also provides a practical way to estimate uncertainty and to work with large or nonlinear models. The sources also note that some modern VI methods can use neural networks or invertible transformations to make the approximation family more flexible.

[2] Bayesian inference in generative models [3] A primer on variational inference for physics-informed deep ...

Limits and caveats

VI is an approximation, not an exact solution. Classical VI may rely on restrictions such as conjugate priors and mean-field assumptions, and some versions struggle to represent multimodal posteriors, meaning distributions with several separated peaks. Other versions require differentiability or reparameterization. The sources also note that uncertainty quantification with VI remains an open practical and theoretical challenge, so the quality of the approximation should be checked rather than assumed.

[2] Bayesian inference in generative models [3] A primer on variational inference for physics-informed deep ... [1] Bayesian Brains and the Rényi Divergence

Sources and further reading

This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.

1. Bayesian Brains and the Rényi Divergence
2. Bayesian inference in generative models
3. A primer on variational inference for physics-informed deep ...