masters-thesis

commit 1f230ab33424712515fca44f99ddf6693a750abe
parent 86bca759bad5355fbe90ef17c716ac352c367225
Author: Silas Brack <s174433@student.dtu.dk>
Date:   Thu,  2 Feb 2023 19:49:09 +0100

Stupid latex.

Diffstat:
M
chapters/training.tex
 | 
4
++--

1 file changed, 2 insertions(+), 2 deletions(-)
diff --git a/chapters/training.tex b/chapters/training.tex
@@ -176,8 +176,8 @@ This means that it is important to compute the negative log-likelihood correctly
 For computing the posterior precision using a mini-batch, we thus have to scale the likelihood term by \(N / B\) to account for the fact that the likelihood is computed on a mini-batch of size \(B\) instead of the full dataset of size \(N\), yielding \(N / B \sum_{i=1}^B \bm J_i \bm H_i \bm J_i + \alpha \identity\).
 We compute the log-determinant of this mini-batched posterior precision as
 \begin{align}
-    \log \det \left( \frac{N}{B} \textstyle{\sum_{i=1}^B} \bm J_i \bm H_i \bm J_i + \alpha \identity \right)
-    ={} & \log \det \left( \frac{N}{B} \left(\textstyle{\sum_{i=1}^B} \bm J_i \bm H_i \bm J_i + \frac{B}{N} \alpha \identity \right) \right)
+    \log\det \left( \frac{N}{B} \textstyle{\sum_{i=1}^B} \bm J_i \bm H_i \bm J_i + \alpha \identity \right)
+    ={} & \log\det \left( \frac{N}{B} \left(\textstyle{\sum_{i=1}^B} \bm J_i \bm H_i \bm J_i + \frac{B}{N} \alpha \identity \right) \right)\hspace{-0.05cm}
     \\ ={} & D \log \frac{N}{B} + \log \det \left( \textstyle{\sum_{i=1}^B} \bm J_i \bm H_i \bm J_i + \frac{B}{N} \alpha \identity \right). \nonumber
 \end{align}
 The scaling is thus important for determining the relative importance of the prior and likelihood terms in the eigenvalues.