Merge pull request #59 from SmirkCao/review01

just update ch10. bjk expression
SmirkCao · Jan 8, 2019 · 2fd4e36 · 2fd4e36
2 parents 84b67db + dfabf85
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diff --git a/.github/ISSUE_TEMPLATE/custom.md b/.github/ISSUE_TEMPLATE/custom.md
diff --git a/APP/README.md b/APP/README.md
diff --git a/CH01/README.md b/CH01/README.md
diff --git a/CH02/Input/data_2-1.txt b/CH02/Input/data_2-1.txt
diff --git a/CH02/Input/latice.xml b/CH02/Input/latice.xml
diff --git a/CH02/Input/logic_data_1.txt b/CH02/Input/logic_data_1.txt
diff --git a/CH02/Input/logic_data_2.txt b/CH02/Input/logic_data_2.txt
diff --git a/CH02/README.md b/CH02/README.md
diff --git a/CH02/assets/XOR_3D.gif b/CH02/assets/XOR_3D.gif
diff --git a/CH03/Input/data_3-1.txt b/CH03/Input/data_3-1.txt
diff --git a/CH03/Input/data_3-2.txt b/CH03/Input/data_3-2.txt
diff --git a/CH03/README.md b/CH03/README.md
diff --git a/CH03/assets/fig3_2.png b/CH03/assets/fig3_2.png
diff --git a/CH03/knn.py b/CH03/knn.py
diff --git a/CH03/unit_test.py b/CH03/unit_test.py
diff --git a/CH04/Input/data_4-1.txt b/CH04/Input/data_4-1.txt
diff --git a/CH04/README.md b/CH04/README.md
diff --git a/CH04/assets/nb_pgm.png b/CH04/assets/nb_pgm.png
diff --git a/CH04/nb.py b/CH04/nb.py
diff --git a/CH04/unit_test.py b/CH04/unit_test.py
diff --git a/CH05/README.md b/CH05/README.md
diff --git a/CH05/assets/Economics_Gini_coefficient2.svg b/CH05/assets/Economics_Gini_coefficient2.svg
diff --git a/CH05/assets/熵与概率的关系.png b/CH05/assets/熵与概率的关系.png
diff --git a/CH05/dt.py b/CH05/dt.py
diff --git a/CH05/unit_test.py b/CH05/unit_test.py
diff --git a/CH06/Input/data.txt b/CH06/Input/data.txt
diff --git a/CH06/README.md b/CH06/README.md
diff --git a/CH06/demo.py b/CH06/demo.py
diff --git a/CH06/maxent.py b/CH06/maxent.py
diff --git a/CH07/README.md b/CH07/README.md
diff --git a/CH07/svm.py b/CH07/svm.py
diff --git a/CH07/unit_test.py b/CH07/unit_test.py
diff --git a/CH09/README.md b/CH09/README.md
@@ -145,12 +145,12 @@ while flag_a or flag_b:
 
 三硬币模型可以写作
 $$\begin{equation}
-        \begin{aligned}
-            P(y|\theta)&=\sum_z P(y,z|\theta) \\
-            &=\sum_z P(z|\theta)P(y|z,\theta) \\
-            &=\pi p^y (1-p)^{1-y} + (1-\pi)q^y(1-q)^{1-y}
-        \end{aligned}
-    \end{equation}
+        \begin{aligned}
+            P(y|\theta)&=\sum_z P(y,z|\theta) \\
+            &=\sum_z P(z|\theta)P(y|z,\theta) \\
+            &=\pi p^y (1-p)^{1-y} + (1-\pi)q^y(1-q)^{1-y}
+        \end{aligned}
+    \end{equation}
 $$
 以上
 
@@ -162,7 +162,6 @@ $$
 观测数据表示为$Y=(Y_1, Y_2, Y_3, \dots, Y_n)^T$, 未观测数据表示为$Z=(Z_1,Z_2, Z_3,\dots, Z_n)^T$, 则观测数据的似然函数为
 
 > 其实觉得这里应该是小写的$y=(y_1,y_2,\dots,y_n), z=(z_1, z_2, \dots,z_n)$
-
 $$
 P(Y|\theta) = \sum\limits_{Z}P(Z|\theta)P(Y|Z,\theta)
 $$
@@ -178,7 +177,6 @@ $$
 $$
 \hat \theta = \arg\max\limits_{\theta}\log P(Y|\theta)
 $$
-
 这个题目的标准答案实际上也是未知的. 因为可能生成这样的观测的假设空间太大.
 
 #### 三硬币模型的EM算法
@@ -195,7 +193,6 @@ EM算法首选参数初值, 记作$\theta^{(0)}=(\pi^{(0)},p^{(0)}, q^{(0)})$,
 $$
 \mu_j^{i+1} = \frac{\pi^{(i)}(p^{(i)})^{y_j}(1-p^{(i)})^{1-y_j}}{\pi^{(i)}(p^{(i)})^{y_j}(1-p^{(i)})^{1-y_j} + (1-\pi^{(i)})(q^{(i)})^{y_j}(1-q^{(i)})^{1-y_j}}
 $$
-
 因为是硬币, 只有0,1两种可能, 所有有上面的表达.
 
 这个表达方式还可以拆成如下形式
@@ -206,7 +203,6 @@ $$
 \frac{\pi^{(i)}(1-p^{(i)})}{\pi^{(i)}(1-p^{(i)}) + (1-\pi^{(i)})(1-q^{(i)})}&, y_j = 0\\
 \end{cases}
 $$
-
 所以, 这步(求$\mu_j$)干了什么, 样本起到了什么作用?
 
  这一步, 通过假设的参数, 计算了不同的样本对假设模型的响应($\mu_j$), 注意这里因为样本($y_j$)是二值的,所以, 用$\{y_j, 1-y_j\}$ 构成了one-hot的编码, 用来表示样本归属的假设.
@@ -216,7 +212,6 @@ $$
 这一步是什么的期望? 书中有写, **观测数据来自硬币$B$的概率, 在二项分布的情况下, 响应度和概率是一个概念. **这个说明, 有助于后面M步公式的理解.
 
 ##### 3.M步
-
 $$
 \begin{align}
 \pi^{(i+1)} &= \frac{1}{n}\sum_{j=1}^{n}\mu_j^{(i+1)}\\
@@ -226,7 +221,6 @@ p^{(i+1)} &= \frac{\sum_{j=1}^{n}\mu_j^{(i+1)}y_j}{\sum_{j=1}^{n}\mu_j^{(i+1)}}\
 q^{(i+1)} &= \frac{\sum_{j=1}^{n}(1-\mu_j^{(i+1)})y_j}{\sum_{j=1}^{n}(1-\mu_j^{(i+1)})}
 \end{align}
 $$
-
 上面, 红色部分的公式从`观测数据是来自硬币B的概率`这句来理解.
 
 ##### 初值影响
@@ -344,11 +338,9 @@ $$
     高斯混合模型的参数估计是EM算法的一个重要应用, 隐马尔科夫模型的非监督学习也是EM算法的一个重要应用.
 
 1. 书中描述的是一维的高斯混合模型, d维的形式如下[^2], 被称作多元正态分布, 也叫多元高斯分布
-
 $$
 \phi(y|\theta_k)=\frac{1}{\sqrt{(2\pi)^d|\Sigma|}}\exp\left(-\frac{(y-\mu_k)^T\Sigma^{-1}(y-\mu_k)}{2}\right)其中,协方差矩阵
 $$
-
 其中,协方差矩阵$\Sigma\in \R^{n\times n}$
 
 #### GMM的图模型
@@ -374,7 +366,6 @@ P(y|\theta)=&\prod_{j=1}^NP(y_j|\theta)\\
 =&\prod_{j=1}^N\sum_{k=1}^K\alpha_k\phi(y|\theta_k)
 \end{align}
 $$
-
 使用EM算法估计GMM的参数$\theta$
 
 ##### 1. 明确隐变量, 初值
@@ -385,16 +376,14 @@ $$
   1. 依第$k$个分模型的概率分布$\phi(y|\theta_k)$生成观测数据$y_j$
   1. 反映观测数据$y_j$来自第$k$个分模型的数据是**未知的**, $k=1,2,\dots,K$ 以**隐变量$\gamma_{jk}$**表示
      **注意这里$\gamma_{jk}$的维度$(j\times k)$**
-
-  $$
+$$
   \gamma_{jk}=
   \begin{cases}
   1, &第j个观测来自第k个分模型\\
   0, &否则
   \end{cases}\\
   j=1,2,\dots,N; k=1,2,\dots,K; \gamma_{jk}\in\{0,1\}
-  $$
-
+$$
   注意, 以上说明有几个假设:
 
   1. 隐变量和观测变量的数据对应, 每个观测数据, 对应了一个隐变量, $\gamma_{jk}$是一种one-hot的形式.
@@ -403,21 +392,20 @@ $$
 - 完全数据为$(y_j,\gamma_{j1},\gamma_{j2},\dots,\gamma_{jK},k=1,2,\dots,N)$
 
 - 完全数据似然函数
-  $$
+$$
   \begin{aligned}
   P(y,\gamma|\theta)=&\prod_{j=1}^NP(y_j,\gamma_{j1},\gamma_{j2},\dots,\gamma_{jK}|\theta)\\
   =&\prod_{k=1}^K\prod_{j=1}^N\left[\alpha_k\phi(y_j|\theta_k)\right]^{\gamma_{jk}}\\
   =&\prod_{k=1}^K\alpha_k^{n_k}\prod_{j=1}^N\left[\phi(y_j|\theta_k)\right]^{\gamma_{jk}}\\
   =&\prod_{k=1}^K\alpha_k^{n_k}\prod_{j=1}^N\left[\frac{1}{\sqrt{2\pi}\sigma_k}\exp\left(-\frac{(y_j-\mu_k)^2}{2\sigma^2}\right)\right]^{\gamma_{jk}}\\
   \end{aligned}
-  $$
+$$
   其中$n_k=\sum_{j=1}^N\gamma_{jk}, \sum_{k=1}^Kn_k=N$
 
 - 完全数据对数似然函数
-  $$
+$$
   \log P(y,\gamma|\theta)=\sum_{k=1}^K\left\{n_k\log \alpha_k+\sum_{j=1}^N\gamma_{jk}\left[\log \left(\frac{1}{\sqrt{2\pi}}\right)-\log \sigma_k -\frac{1}{2\sigma^2}(y_j-\mu_k)^2\right]\right\}
-  $$
-
+$$
 ##### 2. E步,确定Q函数
 
 把$Q$ 函数表示成参数形式

diff --git a/CH09/assets/gmm_graph_model.png b/CH09/assets/gmm_graph_model.png
diff --git a/CH09/bmm.py b/CH09/bmm.py
diff --git a/CH09/gmm.py b/CH09/gmm.py
diff --git a/CH09/model.py b/CH09/model.py
diff --git a/CH09/unit_test.py b/CH09/unit_test.py
diff --git a/CH10/Input/data_10-1.txt b/CH10/Input/data_10-1.txt
diff --git a/CH10/Input/data_10-2.txt b/CH10/Input/data_10-2.txt
diff --git a/CH10/README.md b/CH10/README.md
@@ -247,6 +247,7 @@ $$
 >
 >
 >
+>
 
 - 这里需要**注意**下, 按照后向算法, $\beta$在递推过程中会越来越小, 如果层数较多, 怕是$P(O|\lambda)$会消失
 - 另外一个要注意的点$\color{red}o_{t+1}\beta_{t+1}$
@@ -316,7 +317,23 @@ $$
 
 $\gamma$的维度应该是$N\times T$, 通过$\sum\limits_{t=1}^T$可以降维到$N$, 但是实际上$B$的维度是$N\times M$, 所以有了这个表达, 窃以为这里可以表示成$b_{jk}$, 书中对应部分的表达在$P_{172}的10.3$, 也说明了$b_j(k)$的具体定义. 
 
-这里涉及到实际实现的时候, 可以考虑把观测序列$O$转换成one-hot的形式, $O_{one\_hot}$维度为$M\times T$,$B$的维度$N\times M$,  $B\cdot O$之后, 转换成观测序列对应的发射概率矩阵, 维度为$N\times T$.
+这里涉及到实际实现的时候, 可以考虑把观测序列$O$转换成one-hot的形式, $O_{one\_hot}$维度为$M\times T$,$B$的维度$N\times M$,  $B\cdot O$之后, 转换成观测序列对应的发射概率矩阵, 维度为$N\times T$.
+
+补充一下， $o_t=v_k$有另外一种表达是$ \sigma_{o_t,v_k}$， 克罗内克函数。
+
+克罗内克函数是一个二元函数， 自变量一般是两个整数， 如果两者相等， 输出是1, 否则为0.
+
+其实和指示函数差不多， 只不过条件只限制在了相等。
+$$
+\sigma_{ij}=
+\begin{cases}
+1 (i = j)\\
+0 (i\ne j)
+\end{cases}
+\\
+b_j(k)=\frac{\sum\limits_{t=1,o_t=v_k}^{T}\gamma_t(j)}{\sum\limits_{t=1}^T\gamma_t(j)}=\frac{\sum\limits_{t=1}^{T}\sigma_{o_t,v_k}\gamma_t(j)}{\sum\limits_{t=1}^T\gamma_t(j)}
+$$
+
 
 #### $E$步与$M$步的理解
 

diff --git a/CH10/assets/graph_model.png b/CH10/assets/graph_model.png
diff --git a/CH10/hmm.py b/CH10/hmm.py
diff --git a/CH10/unit_test.py b/CH10/unit_test.py
diff --git a/CH11/Input/template b/CH11/Input/template
diff --git a/CH11/README.md b/CH11/README.md
diff --git a/CH11/assets/1537524145846.png b/CH11/assets/1537524145846.png
diff --git a/CH11/crf.py b/CH11/crf.py
diff --git a/CH11/unit_test.py b/CH11/unit_test.py
diff --git a/CH12/README.md b/CH12/README.md
diff --git a/CH12/assets/loss.png b/CH12/assets/loss.png
diff --git a/README.md b/README.md
diff --git a/Refs/README.md b/Refs/README.md
diff --git a/assets/content_distribution.png b/assets/content_distribution.png
diff --git a/assets/data_algo_map.png b/assets/data_algo_map.png
diff --git a/errata.md b/errata.md
@@ -54,5 +54,5 @@
 
 1. $P_{140}$例题来源于http://www.csie.edu.tw， 这个大概应该是http://www.csie.ntu.edu.tw。 但是也没找到对应的例子页面。
 
-1. 
+1. $P_{148}$在提升树这个地方， 最后得到的提升树是$f_M(x)$， 前面介绍加法模型的时候， 得到的是$f(x)$实际上是一样的意思， 但是两个地方的表达不太一样。这个， 其实不算吧。。
 
diff --git a/glossary_index.md b/glossary_index.md
diff --git a/math_markdown.md b/math_markdown.md
diff --git a/math_markdown.pdf b/math_markdown.pdf
diff --git a/notebook/README.md b/notebook/README.md
diff --git a/ref_downloader.sh b/ref_downloader.sh
diff --git a/requirements.txt b/requirements.txt