Laplace's approximation
Laplace's approximation provides an analytical expression for a posterior probability distribution by fitting a Gaussian distribution with a mean equal to the MAP solution and precision equal to the observed Fisher information. The approximation is justified by the Bernstein–von Mises theorem, which states that under regularity conditions the posterior converges to a Gaussian in large samples. For example, a (possibly non-linear) regression or classification model with data set  comprising inputs  and outputs  has (unknown) parameter vector  of length . The likelihood is denoted  and the parameter prior . The joint density of outputs and parameters  is the object of inferential desire

The joint is equal to the product of the likelihood and the prior and by Bayes' rule, equal to the product of the marginal likelihood  and posterior . Seen as a function of  the joint is an un-normalised density. In Laplace's approximation we approximate the joint by an un-normalised Gaussian , where we use  to denote approximate density,  for un-normalised density and  is a constant (independent of ). Since the marginal likelihood  doesn't depend on the parameter  and the posterior  normalises over  we can immediately identify them with  and  of our approximation, respectively. Laplace's approximation is

where we have defined

where  is the location of a mode of the joint target density, also known as the maximum a posteriori or MAP point and  is the  positive definite matrix of second derivatives of the negative log joint target density at the mode . Thus, the Gaussian approximation matches the value and the curvature of the un-normalised target density at the mode. The value of  is usually found using a gradient based method, e.g. Newton's method. In summary, we have

for the approximate posterior over  and the approximate log marginal likelihood respectively. In the special case of Bayesian linear regression with a Gaussian prior, the approximation is exact. The main weaknesses of Laplace's approximation are that it is symmetric around the mode and that it is very local: the entire approximation is derived from properties at a single point of the target density. Laplace's method is widely used and was pioneered in the context of neural networks by David MacKay, and for Gaussian processes by Williams and Barber. References 

Statistical approximations
Bayesian inference