Laplace Approximation

TL;DR

• Approximates a complex posterior with a Gaussian to make probabilities and inference easier.
• Enables more efficient optimization of model parameters.
• Commonly applied in settings like Bayesian linear regression and variational inference.

Definition

Laplace approximation is a method used in statistics and machine learning to approximate a complex posterior distribution with a simpler, Gaussian distribution, allowing easier calculation of probabilities and inference and more efficient optimization of model parameters.

Explanation

The Laplace approximation replaces a difficult-to-handle posterior distribution by a Gaussian approximation, which simplifies analytical and computational tasks. By using a Gaussian surrogate, one can perform probability calculations and inference more straightforwardly and optimize model parameters more efficiently than working with the original complex posterior.

Examples

Bayesian linear regression

In Bayesian linear regression, the posterior distribution of model parameters is often complex and difficult to work with. Using Laplace approximation, this posterior can be approximated as a Gaussian distribution, allowing for easier calculation of probabilities and inference.

Variational inference

In variational inference, Laplace approximation is used to approximate a complex posterior distribution with a simpler distribution, such as a Gaussian. This facilitates more efficient optimization of model parameters and improved inference.

Use cases

• Easier calculation of probabilities and inference when the true posterior is complex.
• More efficient optimization of model parameters by working with a Gaussian approximation.

Related terms

• Gaussian distribution
• Posterior distribution
• Bayesian linear regression
• Variational inference