Merge branch 'TeXNotes' of github.com:PhotonYan/MachineLearningCourse2025 into TeXNotes

Author: PhotonYan

README.md

@@ -23,3 +23,7 @@ https://disk.pku.edu.cn/link/AA61A92A21D0404370B153E970F17D6F6F
 2025.09.17 【通知】第一次课后练习已发布，DDL为10.1（周三）晚11:59，请大家按时完成
 
 2025.10.15 【通知】课后练习2、3已经发布，截止时间为10.29晚11:59，注意一次作业有四道题。**为方便助教批改以及给大家反馈，请大家提交一个【pdf文件】，不要直接提交多张图片或者提交word**
+
+**期末考试信息：
+2026年1月7日 周三 下午14:00-16:00 一教101
+请大家按时参加！注意cheating sheet需要纸笔手写，一页A4正反面，不能打印平板，到时助教会检查。**

notes/2024/generative-model.md

@@ -339,7 +339,7 @@ $$
 $$
 \begin{aligned}
 \mathrm{ELBO} &= \log p_{\theta}(x_0) - \mathrm{KL} \\
-&= \int_{x_{1:T}}q(x_{1:T}\mid x_0) \log p_{\theta}(x_0) \mathrm{d}x_{1:T} - \int_{x_{1:T}} q(x_{1:T}\mid x_0) \frac{q(x_{1:T}\mid x_0)}{p_{\theta}(x_{1:T}\mid x_0)} \mathrm{d} x_{1:T}\\
+&= \int_{x_{1:T}}q(x_{1:T}\mid x_0) \log p_{\theta}(x_0) \mathrm{d}x_{1:T} - \int_{x_{1:T}} q(x_{1:T}\mid x_0) \log \frac{q(x_{1:T}\mid x_0)}{p_{\theta}(x_{1:T}\mid x_0)} \mathrm{d} x_{1:T}\\
 &= \int_{x_{1:T}} q(x_{1:T}\mid x_0) \log \frac{p_{\theta}(x_{0:T})}{q(x_{1:T}\mid x_0)}\mathrm{d} x_{1:T}\\
 &= \int_{x_{1:T}} q(x_{1:T}\mid x_0) \log \frac{p(x_T) \prod_{t \in [T]} p_{\theta}(x_{t-1}\mid x_t) }{\prod_{t \in [T]} q(x_t\mid x_{t-1}) }\mathrm{d} x_{1:T}\\
 &= \mathbb{E}_{x_{1:T} \sim  q(x_{1:T}\mid x_0)}\left[ \log p(x_T) + \sum_{t \in [T]}\log  \frac{p_{\theta}(x_{t-1}\mid x_t)}{q(x_t\mid x_{t-1})} \right] 
@@ -449,7 +449,7 @@ $$
 然后把 $q(x_{t-1},x_t\mid x_0)$ 拆成 $q(x_{t-1}\mid x_t,x_0)\cdot q(x_t\mid x_0)$，并将对 $x_t$ 的积分拿到最外面：
 
 $$
-\int_{x_t} q(x_t\mid x_0)\int_{x_{t-1}} q(x_{t-1}\mid x_t,x_0) \log \frac{q(x_{t-1}\mid x_t,x_0)}{p_{\theta}(x_{t-1},x_t)}\mathrm{d}x_{t-1}\mathrm{d}x_t
+\int_{x_t} q(x_t\mid x_0)\int_{x_{t-1}} q(x_{t-1}\mid x_t,x_0) \log \frac{q(x_{t-1}\mid x_t,x_0)}{p_{\theta}(x_{t-1} \mid x_t)}\mathrm{d}x_{t-1}\mathrm{d}x_t
 $$
 
 KL 散度的形式这就出来了！