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

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