Laplace's approximation

Not to be confused with Laplace's method, which is based on an essentially identical construction. Whereas Laplace's method focusses on a limiting behaviour of the integral, Laplace's approximation isn't used in the limit, and considers both integral and integrand. This naming distinction may not be universal.

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.^[1][2] The approximation is justified by the Bernstein–von Mises theorem, which states that, under regularity conditions, the error of the approximation tends to 0 as the number of data points tends to infinity.^[3][4]

For example, consider a regression or classification model with data set ${\displaystyle \{x_{n},y_{n}\}_{n=1,\ldots ,N}}$ comprising inputs ${\displaystyle x}$ and outputs ${\displaystyle y}$ with (unknown) parameter vector ${\displaystyle \theta }$ of length ${\displaystyle D}$. The likelihood is denoted ${\displaystyle p({\bf {y}}|{\bf {x}},\theta )}$ and the parameter prior ${\displaystyle p(\theta )}$. Suppose one wants to approximate the joint density of outputs and parameters ${\displaystyle p({\bf {y}},\theta |{\bf {x}})}$. Bayes' formula reads:

${\displaystyle p({\bf {y}},\theta |{\bf {x}})\;=\;p({\bf {y}}|{\bf {x}},\theta )p(\theta |{\bf {x}})\;=\;p({\bf {y}}|{\bf {x}})p(\theta |{\bf {y}},{\bf {x}})\;\simeq \;{\tilde {q}}(\theta )\;=\;Zq(\theta ).}$

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 ${\displaystyle p({\bf {y}}|{\bf {x}})}$ and posterior ${\displaystyle p(\theta |{\bf {y}},{\bf {x}})}$. Seen as a function of ${\displaystyle \theta }$ the joint is an un-normalised density.

In Laplace's approximation, we approximate the joint by an un-normalised Gaussian ${\displaystyle {\tilde {q}}(\theta )=Zq(\theta )}$, where we use ${\displaystyle q}$ to denote approximate density, ${\displaystyle {\tilde {q}}}$ for un-normalised density and ${\displaystyle Z}$ the normalisation constant of ${\displaystyle {\tilde {q}}}$ (independent of ${\displaystyle \theta }$). Since the marginal likelihood ${\displaystyle p({\bf {y}}|{\bf {x}})}$ doesn't depend on the parameter ${\displaystyle \theta }$ and the posterior ${\displaystyle p(\theta |{\bf {y}},{\bf {x}})}$ normalises over ${\displaystyle \theta }$ we can immediately identify them with ${\displaystyle Z}$ and ${\displaystyle q(\theta )}$ of our approximation, respectively.

Laplace's approximation is

${\displaystyle p({\bf {y}},\theta |{\bf {x}})\;\simeq \;p({\bf {y}},{\hat {\theta }}|{\bf {x}})\exp {\big (}-{\tfrac {1}{2}}(\theta -{\hat {\theta }})^{\top }S^{-1}(\theta -{\hat {\theta }}){\big )}\;=\;{\tilde {q}}(\theta ),}$

where we have defined

${\displaystyle {\begin{aligned}{\hat {\theta }}&\;=\;\operatorname {argmax} _{\theta }\log p({\bf {y}},\theta |{\bf {x}}),\\S^{-1}&\;=\;-\left.\nabla _{\theta }\nabla _{\theta }\log p({\bf {y}},\theta |{\bf {x}})\right|_{\theta ={\hat {\theta }}},\end{aligned}}}$

where ${\displaystyle {\hat {\theta }}}$ is the location of a mode of the joint target density, also known as the maximum a posteriori or MAP point and ${\displaystyle S^{-1}}$ is the ${\displaystyle D\times D}$ positive definite matrix of second derivatives of the negative log joint target density at the mode ${\displaystyle \theta ={\hat {\theta }}}$. Thus, the Gaussian approximation matches the value and the log-curvature of the un-normalised target density at the mode. The value of ${\displaystyle {\hat {\theta }}}$ is usually found using a gradient based method.

In summary, we have

${\displaystyle {\begin{aligned}q(\theta )&\;=\;{\cal {N}}(\theta |\mu ={\hat {\theta }},\Sigma =S),\\\log Z&\;=\;\log p({\bf {y}},{\hat {\theta }}|{\bf {x}})+{\tfrac {1}{2}}\log |S|+{\tfrac {D}{2}}\log(2\pi ),\end{aligned}}}$

for the approximate posterior over ${\displaystyle \theta }$ and the approximate log marginal likelihood respectively.

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,^[5] and for Gaussian processes by Williams and Barber.^[6]