Laplace Approximation

The likelihood function is the joint density of the observations i.e.

\[ L(\theta) = f(Y) \] Now consider instead the joint density of both states \(X\) and observations \(Y\). This can be written as

\[ f(Y) = \int_{X} f(X,Y) \, \mathrm{d}X = \int_{X} \exp \log f(X,Y) \, \mathrm{d}X \]