conclusion of bayesian stuff

1 files changed, 16 insertions, 3 deletions

diff --git a/finale/sections/bayesian.tex b/finale/sections/bayesian.tex
index 1426c97..6e44b08 100644
--- a/finale/sections/bayesian.tex
+++ b/finale/sections/bayesian.tex
@@ -119,6 +119,19 @@ counterpart.
   \end{split}
 \end{equation}
 
-Reparametrization trick
-Batches
-Algorithm
+Once again, the product form of the prior allows us to decompose the expected
+log-likelihood term as such:
+\begin{equation}
+  \begin{split}
+    &\mathbb{E}_{q_{\mathbf{\Theta'}}} \log \mathcal{L}(\mathbf{x}_c |
+    \mathbf{\Theta}) = - \sum_{j,c , t } \mu_j \cdot x^t_c \\
+    &+ \sum_{j, c, t} \mathbb{E}_{\theta_j \sim\mathcal{N}(\vec 0_n, I_n)}
+    \left[ (x_c^{t+1})_j \log \left(1 - e^{-(\sigma_j \cdot \theta_j + \mu_j)
+    \cdot x_c^t}\right) \right]
+  \end{split}
+\end{equation}
+
+Note that by reparametrizing $\theta_j' = \sigma_j \cdot \theta_j + \mu_j$,
+where $\sigma_j \cdot \theta_j$ is the elementwise multiplication operation, we
+can easily compute the gradient of the variational objective with respect to
+$\mu_j$ and $\sigma_j$.