Laplace's Method

Suppose we have an unnormalized probability density $$P^\star(x)$$, whose normalizing constant:

$$Z_P \equiv \int P^\star(x) dx$$

is of interest, and has a peak at point $$x_0$$.

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

$$\ln P^\star(x) \approxeq \ln P^\star(x_0) - \frac{c}{2}(x - x_0)^2 + \dots$$

where $$c = - \frac{\partial^2}{\partial x^2} \ln P^\star(x) \text{ where } {x = x_0}$$.

$$P^\star(x)$$ can be approximated by an unnormalized Gaussian:

$$Q^\star(x) \equiv P^\star(x_0) \textrm{exp}\left[- \frac{c}{2} (x-x_0)^2\right]$$

and the normalizing constant is approximated with:

$$Z_Q \equiv P^\star(x_0) \sqrt{\frac{2\pi}{c}}$$

This is easily generalizable to a K-dimensional space $$\mathbf{x}$$.