# Laplace Approximation

## Definition of Laplace Approximation

Laplace Approximation: The Laplace approximation is a method used in mathematics to approximate the value of a function. It is named after the mathematician Pierre-Simon Laplace, who first proposed it in 1774. The approximation is based on the assumption that the function is smooth, which means that it can be approximated by a polynomial.

## How is Laplace Approximation used?

Laplace Approximation is an important tool in Bayesian data analysis and machine learning. It relies on approximating a complex function by a simple one, typically a Gaussian distribution. This approximation is based on the assumption that there is a linear relationship between the two functions, and that the resulting approximation should be as close to the original function as possible. The Laplace Approximation is usually used when dealing with high-dimensional probability distributions or posterior densities, because it allows us to estimate the integral of these distributions quickly and efficiently.

In Bayesian data analysis, Laplace Approximation is used to calculate the posterior density (the probability of different outcomes given a set of data or observations). In this context, it enables us to easily compute marginal likelihoods for different models and parameters, which can then be compared to determine which model is most likely given our data. In machine learning, Laplace Approximation can be used in algorithms such as support vector machines and neural networks by providing more efficient estimates of various hyperparameters such as learning rates and regularization constants.

The central idea behind Laplace Approximation is that we can approximate a complex function by selecting an appropriate family of simpler functions (usually Gaussian functions). We then use numerical methods to compute the best fit parameters for this family of functions relative to our original complex function. This fitting process then produces an approximation of our original function which is both fast to calculate with good accuracy.