---
title: "Laplace Approximation"
author: ""
date: ""
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Laplace Approximation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

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

