aner/thesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Before the big switch.

Browse Source
This commit is contained in:
Aner Zakobar
2026-07-02 22:54:10 +03:00
parent 7ff2ef6835
commit 10f1bd53f1
1 changed files with 5 additions and 5 deletions
Download Patch File Download Diff File Expand all files Collapse all files
content/method/method.tex
+5 -5
View File
@@ -526,7 +526,7 @@ Firstly, for the true population, we calculate the derivatives.
\[
\dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial f_{\theta}(\mathbf{x}_i)} =
\frac{\partial \sum_{\mathbf{x}_j \in \mathcal{B}_1} (\sigma_1(f_{\theta}(\mathbf{x}_j) - \tau))}{\partial f_{\theta}(\mathbf{x}_i)}
\frac{\partial \frac{1}{|\mathcal{B}_1|}\sum_{\mathbf{x}_j \in \mathcal{B}_1} (\sigma_1(f_{\theta}(\mathbf{x}_j) - \tau))}{\partial f_{\theta}(\mathbf{x}_i)}
\]
\[
@@ -541,11 +541,11 @@ Now, for the false population. As stated in the roll framework equation above (T
\[
\frac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \hat{F}_0^{-1}(\alpha)} = \frac{\partial \sum_{\mathbf{x}_j \in \mathcal{B}_1} (\sigma_1(f_{\theta}(\mathbf{x}_j) - \tau))}{\partial \tau}
\frac{\partial \hat{F}_1(\hat{F}_0^{-1}(\alpha))}{\partial \hat{F}_0^{-1}(\alpha)} = \frac{\partial \frac{1}{|\mathcal{B}_1|}\sum_{\mathbf{x}_j \in \mathcal{B}_1} (\sigma_1(f_{\theta}(\mathbf{x}_j) - \tau))}{\partial \tau}
\]
\[
= -\frac{1}{|\mathcal{B}_1|}\sum_{\mathbf{x}_i \in \mathcal{B}_1}\sigma_1'(f_{\theta}(\mathbf{x}_i) - \tau)
= -\frac{1}{|\mathcal{B}_1|}\sum_{\mathbf{x}_j \in \mathcal{B}_1}\sigma_1'(f_{\theta}(\mathbf{x}_j) - \tau)
\]
At this point, we must calculate the derivative of the threshold w.r.t each score from the false population.
@@ -569,13 +569,13 @@ Therefore, we must invert the function in order to calculate $\mathbf{x}_i$ give
This inverse equation is given by:
\[
\sigma^{-1}\left( |\mathcal{B}_0|\cdot\alpha - \sum_{\mathbf{x}_j \in \mathcal{B}_0 ; j \neq i}\sigma(\tau - f_{\theta}(\mathbf{x}_j))\right) + \tau
\tau - \sigma^{-1}\left( |\mathcal{B}_0|\cdot\alpha - \sum_{\mathbf{x}_j \in \mathcal{B}_0 ; j \neq i}\sigma(\tau - f_{\theta}(\mathbf{x}_j))\right)
\]
Applying the inverse derivative rule to this inverse:
\[
\frac{\partial \hat{F}_0^{-1}(\alpha ; \mathcal{B}_0)}{\partial f_\theta(\mathbf{x}_i)} = \frac{1}{\frac{\partial \sigma^{-1}\left( |\mathcal{B}_0|\cdot \alpha - \sum_{\mathbf{x}_j \in \mathcal{B}_0 ; j \neq i}\sigma(\tau - f_{\theta}(\mathbf{x}_j))\right) + \tau}{\partial \tau}}
\frac{\partial \hat{F}_0^{-1}(\alpha ; \mathcal{B}_0)}{\partial f_\theta(\mathbf{x}_i)} = \frac{1}{\frac{\partial \left(\tau - \sigma^{-1}\left( |\mathcal{B}_0|\cdot \alpha - \sum_{\mathbf{x}_j \in \mathcal{B}_0 ; j \neq i}\sigma(\tau - f_{\theta}(\mathbf{x}_j))\right)\right)}{\partial \tau}}
\]
Applying the chain rule: