Laplace's approximation

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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]