add details on how laplace approximation provides approximation for marginal likelihood.

Author: charlesm93

src/functions-reference/embedded_laplace.qmd

@@ -31,7 +31,7 @@ a two-step procedure:
 In the above procedure, neither the marginal posterior nor the conditional posterior
 are typically available in closed form and so they must be approximated.
 The marginal posterior can be written as  $p(\phi \mid y) \propto p(y \mid \phi) p(\phi)$,
-where  $p(y \mid \phi) = \int p(y \mid \phi, \theta) p(\theta) \text{d}\theta$ 
+where  $p(y \mid \phi) = \int p(y \mid \phi, \theta) p(\theta) \text{d}\theta$
 is called the marginal likelihood.  The Laplace method approximates
 $p(y \mid \phi, \theta) p(\theta)$ with a normal distribution centered at
 $$
@@ -41,6 +41,10 @@ and $\theta^*$ is obtained using a numerical optimizer.
 The resulting Gaussian integral can be evaluated analytically to obtain an
 approximation to the log marginal likelihood
 $\log \hat p(y \mid \phi) \approx \log p(y \mid \phi)$.
+Specifically:
+$$
+  \hat p(y \mid \phi) = \frac{p(\theta^* \mid \phi) p(y \mid \theta^*, \phi)}{\hat p (\theta^* \mid \phi, y)}.
+$$
 
 Combining this marginal likelihood with the prior in the `model`
 block, we can then sample from the marginal posterior $p(\phi \mid y)$