## Laplace's method

Laplace's method is a standard procedure for approximating expectations over posterior pdf (probability density function). Here we only consider approximating the posterior pdf.

The basic idea is to apply Taylor's series approximation for the logarithm of the posterior pdf. In practise, the second order approximation is used. This amounts to approximating the posterior pdf by the Gaussian distribution.

Let us denote the posterior pdf by P(x). In practise the procedure is as follows:

• Find the maximum of -log P(x). Denote the maximum point by x0.
• Find the second order Taylor's series approximation of -log P(x) around x0, i.e., compute the the Hessian matrix of -log P(x) at x0
• Normalise the approximated pdf so that it integrates to unity.
If the posterior pdf has several local maxima, x1, x2, ..., xn, one can approximate each of them locally by the Taylor's series and then assume that the whole posterior pdf is a sum of the local approximations.

Back to the exercises.

Tästä sivusta vastaa Harri.Lappalainen@hut.fi.
Sivua on viimeksi päivitetty 4.2.1999.
URL: http://www.cis.hut.fi/cis/edu/Tik-61.181/laplace.html