Laplace's Method

Suppose we have an unnormalized probability density $P^{⋆} (x)$ , whose normalizing constant:

$Z_{P} \equiv \int P^{⋆} (x) d x$

is of interest, and has a peak at point $x_{0}$ .

We perform a Taylor expansion of $\ln P^{⋆} (x)$ at this peak:

$\ln P^{⋆} (x) ≊ \ln P^{⋆} (x_{0}) - \frac{c}{2} (x - x_{0})^{2} + \dots$

where $c = - \frac{\partial^{2}}{\partial x^{2}} \ln P^{⋆} (x) where x = x_{0}$ .

$P^{⋆} (x)$ can be approximated by an unnormalized Gaussian:

$Q^{⋆} (x) \equiv P^{⋆} (x_{0}) exp [- \frac{c}{2} (x - x_{0})^{2}]$

and the normalizing constant is approximated with:

$Z_{Q} \equiv P^{⋆} (x_{0}) \sqrt{\frac{2 π}{c}}$

This is easily generalizable to a K-dimensional space $x$ .