bug fixes (sec 3)

Section03-Word-Based-Models/section03.tex

@@ -909,12 +909,12 @@
 
 \begin{itemize}
 
-\item 很多时候，我们有多个互译句对$(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[n]},\mathbf{t}^{[n]})$，称之为\alert{双语平行数据(语料)}。翻译概率可以被定义为
+\item 如果有多个互译句对$\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[K]},\mathbf{t}^{[K]})\}$，称之为\alert{双语平行数据(语料)}。翻译概率可以被定义为
 
 \vspace{-1em}
 
 \begin{eqnarray}
-\textrm{P}(x,y) & = & \frac{\sum_{i=1}^{n}c(x,y;\mathbf{s}^{[i]},\mathbf{t}^{[i]})}{\sum_{i=1}^{n} \sum_{x',y'} c(x',y';\mathbf{s}^{[i]},\mathbf{t}^{[i]})} \nonumber
+\textrm{P}(x,y) & = & \frac{\sum_{k=1}^{K}c(x,y;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}{\sum_{k=1}^{K} \sum_{x',y'} c(x',y';\mathbf{s}^{[k]},\mathbf{t}^{[k]})} \nonumber
 \end{eqnarray}
 
 \item<2-> 说白了就是计算$(x,y)$的频次时，在每个句子上累加
@@ -1414,7 +1414,7 @@ $m$ & $n$ & $n^m \cdot m!$ \\ \hline
 \node [anchor=north west,inner sep=2pt,align=left] (line4) at ([yshift=-1pt]line3.south west) {\textrm{3: \textbf{for} $i$ in $[1,m]$ \textbf{do}}};
 \node [anchor=north west,inner sep=2pt,align=left] (line5) at ([yshift=-1pt]line4.south west) {\textrm{4: \hspace{1em} $h = \phi$}};
 \node [anchor=north west,inner sep=2pt,align=left] (line6) at ([yshift=-1pt]line5.south west) {\textrm{5: \hspace{1em} \textbf{foreach} $j$ in $[1,m]$ \textbf{do}}};
-\node [anchor=north west,inner sep=2pt,align=left] (line7) at ([yshift=-1pt]line6.south west) {\textrm{6: \hspace{2em} \textbf{if} $used[j]=$ \textbf{true} \textbf{then}}};
+\node [anchor=north west,inner sep=2pt,align=left] (line7) at ([yshift=-1pt]line6.south west) {\textrm{6: \hspace{2em} \textbf{if} $used[j]=$ \textbf{false} \textbf{then}}};
 \node [anchor=north west,inner sep=2pt,align=left] (line8) at ([yshift=-1pt]line7.south west) {\textrm{7: \hspace{3em} $h = h \cup \textrm{\textsc{Join}}(best,\pi[j])$}};
 \node [anchor=north west,inner sep=2pt,align=left] (line9) at ([yshift=-1pt]line8.south west) {\textrm{8: \hspace{1em} $best = \textrm{\textsc{PruneForTop1}}(h)$}};
 \node [anchor=north west,inner sep=2pt,align=left] (line10) at ([yshift=-1pt]line9.south west) {\textrm{9: \hspace{1em} $used[best.j] = \textrm{\textsc{\textbf{true}}}$}};
@@ -2395,7 +2395,7 @@ $m$ & $n$ & $n^m \cdot m!$ \\ \hline
     \item \textbf{翻译模型参数估计} - 计算$\textrm{P}(\mathbf{s}|\mathbf{t})$所需的参数
     \end{itemize}
     \vspace{0.5em}
-\item<2-> \textbf{IBM模型的假设}：$\mathbf{s}=s_1...s_m$和$\mathbf{t}=t_1...t_n$之间有单词一级的对应，称作\alert{单词对齐}或者\alert{词对齐}。此外：
+\item<2-> \textbf{IBM模型的假设}：$\mathbf{s}=s_1...s_m$和$\mathbf{t}=t_1...t_l$之间有单词一级的对应，称作\alert{单词对齐}或者\alert{词对齐}。此外：
     \begin{itemize}
     \item \textbf{约束}：一个源语言单词只能对应一个目标语单词
     \vspace{0.5em}
@@ -2792,11 +2792,11 @@ $\mathbf{s}$ = 在 桌子 上 \ \ \ \ \ $\mathbf{t}$ = $t_0$ on the table \ \ \
 \textrm{P}(\mathbf{s},\mathbf{a}|\mathbf{t}) & = & \textrm{P}(m|\mathbf{t}) \prod\limits_{j=1}^{m} \textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,\mathbf{t}) \textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,\mathbf{t}) \nonumber \\
                   & \visible<2->{=} & \visible<2->{\textrm{P}(m=3 \mid \textrm{'$t_0$ on the table'})} \visible<3->{\times} \nonumber \\
                   &   & \visible<3->{\textrm{P}(a_1=0 \mid \phi,\phi,3,\textrm{'$t_0$ on the table'})} \visible<4->{\times} \nonumber \\
-                  &   & \visible<4->{\textrm{P}(f_1=\textrm{在} \mid \textrm{\{1-0\}},\phi,3,\textrm{'$t_0$ on the table'})} \visible<5->{\times} \nonumber \\
+                  &   & \visible<4->{\textrm{P}(s_1=\textrm{在} \mid \textrm{\{1-0\}},\phi,3,\textrm{'$t_0$ on the table'})} \visible<5->{\times} \nonumber \\
                   &   & \visible<5->{\textrm{P}(a_2=3 \mid \textrm{\{1-0\}},\textrm{'在'},3,\textrm{'$t_0$ on the table'})} \visible<6->{\times} \nonumber \\
-                  &   & \visible<6->{\textrm{P}(f_2=\textrm{桌子} \mid \textrm{\{1-0,2-3\}},\textrm{'在'},3,\textrm{'$t_0$ on the table'})} \visible<7->{\times} \nonumber \\
+                  &   & \visible<6->{\textrm{P}(s_2=\textrm{桌子} \mid \textrm{\{1-0,2-3\}},\textrm{'在'},3,\textrm{'$t_0$ on the table'})} \visible<7->{\times} \nonumber \\
                   &   & \visible<7->{\textrm{P}(a_3=1 \mid \textrm{\{1-0,2-3\}},\textrm{'在 桌子'},3,\textrm{'$t_0$ on the table'})} \visible<8->{\times} \nonumber \\
-                  &   & \visible<8->{\textrm{P}(f_3=\textrm{上} \mid \textrm{\{1-0,2-3,3-1\}},\textrm{'在 桌子'},3,\textrm{'$t_0$ on the table'})} \nonumber
+                  &   & \visible<8->{\textrm{P}(s_3=\textrm{上} \mid \textrm{\{1-0,2-3,3-1\}},\textrm{'在 桌子'},3,\textrm{'$t_0$ on the table'})} \nonumber
 \end{eqnarray}
 }
 
@@ -3730,7 +3730,7 @@ $\mathbf{s}$ = 在 桌子 上 \ \ \ \ \ $\mathbf{t}$ = $t_0$ on the table \ \ \
 
     {\small
     \begin{eqnarray}
-    L(f,\lambda) & = & \frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} \prod\limits_{j=1}^{m} f(s_j|t_i) - \nonumber \\
+    L(f,\lambda) & = & \frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) - \nonumber \\
                  &   & \sum_{t_y} \lambda_{t_y} (\sum_{s_x} f(s_x|t_y) -1) \nonumber
     \end{eqnarray}
     }
@@ -4190,9 +4190,9 @@ f(s_u|t_v) & = & \lambda_{t_v}^{-1} \cdot \textrm{P}(\mathbf{s}|\mathbf{t}) \cdo
 %%% scale it up to the full corpus
 \begin{frame}{在整个数据集上计算}
 \begin{itemize}
-\item \textbf{更真实的情况}：我们拥有一系列互译的句对（称作\alert{平行语料}），记为$\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),(\mathbf{s}^{[2]},\mathbf{t}^{[2]}),...,(\mathbf{s}^{[N]},\mathbf{t}^{[N]})\}$。对于这$N$个训练用句对，定义$f(s_u|t_v)$的期望频次为
+\item \textbf{更真实的情况}：我们拥有一系列互译的句对（称作\alert{平行语料}），记为$\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),(\mathbf{s}^{[2]},\mathbf{t}^{[2]}),...,(\mathbf{s}^{[K]},\mathbf{t}^{[K]})\}$。对于这$K$个训练用句对，定义$f(s_u|t_v)$的期望频次为
     \begin{displaymath}
-    c_{\mathbb{E}}(s_u|t_v) = \sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[i]},\mathbf{t}^{[i]})
+    c_{\mathbb{E}}(s_u|t_v) = \sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})
     \end{displaymath}
 \item<2-> \textbf{于是}
     \begin{center}
@@ -4200,8 +4200,8 @@ f(s_u|t_v) & = & \lambda_{t_v}^{-1} \cdot \textrm{P}(\mathbf{s}|\mathbf{t}) \cdo
     \node [anchor=west,inner sep=2pt] (eq1) at (0,0) {$f(s_u|t_v)$};
     \node [anchor=west] (eq2) at (eq1.east) {$=$\ };
     \draw [-] ([xshift=0.3em]eq2.east) -- ([xshift=11.6em]eq2.east);
-    \node [anchor=south west] (eq3) at ([xshift=1em]eq2.east) {$\sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[i]},\mathbf{t}^{[i]})$};
-    \node [anchor=north west] (eq4) at (eq2.east) {$\sum_{s_u} \sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[i]},\mathbf{t}^{[i]})$};
+    \node [anchor=south west] (eq3) at ([xshift=1em]eq2.east) {$\sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})$};
+    \node [anchor=north west] (eq4) at (eq2.east) {$\sum_{s_u} \sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})$};
 
     \visible<4->{
     \node [anchor=south] (label1) at ([yshift=-6em,xshift=3em]eq1.north west) {利用这个公式计算};
@@ -4250,16 +4250,16 @@ f(s_u|t_v) & = & \lambda_{t_v}^{-1} \cdot \textrm{P}(\mathbf{s}|\mathbf{t}) \cdo
 \label{ibmtraining}
 
 \begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true]{IBM模型1的训练（EM算法）}
-输入: 平行语料$\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[N]},\mathbf{t}^{[N]})\}$\\
+输入: 平行语料$\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[K]},\mathbf{t}^{[K]})\}$\\
 输出：参数$f(\cdot|\cdot)$的最优值\\
-1: \textbf{Function} \textsc{TrainItWithEM}($\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[N]},\mathbf{t}^{[N]})\}$) \\
+1: \textbf{Function} \textsc{TrainItWithEM}($\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[K]},\mathbf{t}^{[K]})\}$) \\
 2: \ \ Initialize $f(\cdot|\cdot)$ \hspace{5em} $\rhd$ 比如给$f(\cdot|\cdot)$一个均匀分布\\
 3: \ \ Loop until $f(\cdot|\cdot)$ converges\\
-4: \ \ \ \ \textbf{foreach} $k = 1$ to $N$ \textbf{do}\\
+4: \ \ \ \ \textbf{foreach} $k = 1$ to $K$ \textbf{do}\\
 5: \ \ \ \ \ \ \ \footnotesize{$c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]}) = \sum\limits_{j=1}^{|\mathbf{s}^{[k]}|} \delta(s_j,s_u) \sum\limits_{i=0}^{|\mathbf{t}^{[k]}|} \delta(t_i,t_v) \cdot \frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}\normalsize{}\\
-6: \ \ \ \ \textbf{foreach} $t_v$ appears at least one of $\{\mathbf{t}^{[1]},...,\mathbf{t}^{[N]}\}$ \textbf{do}\\
+6: \ \ \ \ \textbf{foreach} $t_v$ appears at least one of $\{\mathbf{t}^{[1]},...,\mathbf{t}^{[K]}\}$ \textbf{do}\\
 7: \ \ \ \ \ \ \ $\lambda_{t_v}^{'} = \sum_{s_u} \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})$\\
-8: \ \ \ \ \ \ \ \textbf{foreach} $s_u$ appears at least one of $\{\mathbf{s}^{[1]},...,\mathbf{s}^{[N]}\}$ \textbf{do}\\
+8: \ \ \ \ \ \ \ \textbf{foreach} $s_u$ appears at least one of $\{\mathbf{s}^{[1]},...,\mathbf{s}^{[K]}\}$ \textbf{do}\\
 9: \ \ \ \ \ \ \ \ \ $f(s_u|t_v) = \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]}) \cdot (\lambda_{t_v}^{'})^{-1}$\\
 10: \ \textbf{return} $f(\cdot|\cdot)$
 \end{beamerboxesrounded}
@@ -4287,8 +4287,8 @@ c_{\mathbb{E}}(i|j,m,l;\mathbf{s},\mathbf{t}) & = & \frac{f(s_j|t_i)a(i|j,m,l)}{
 \end{eqnarray}
 \item \textbf{M-Step}
 \begin{eqnarray}
-f(s_u|t_v) & = & \frac{\sum_{k=0}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}{\sum_{s_u} \sum_{k=0}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}  \nonumber \\
-a(i|j,m,l) & = & \frac{\sum_{k=0}^{K} c_{\mathbb{E}}(i|j;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}{\sum_{i} \sum_{k=0}^{K} c_{\mathbb{E}}(i|j;\mathbf{s}^{[k]},\mathbf{t}^{[k]})} \nonumber
+f(s_u|t_v) & = & \frac{\sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}{\sum_{s_u} \sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}  \nonumber \\
+a(i|j,m,l) & = & \frac{\sum_{k=1}^{K} c_{\mathbb{E}}(i|j;\mathbf{s}^{[k]},\mathbf{t}^{[k]})}{\sum_{i} \sum_{k=1}^{K} c_{\mathbb{E}}(i|j;\mathbf{s}^{[k]},\mathbf{t}^{[k]})} \nonumber
 \end{eqnarray}
 \end{enumerate}
 \end{frame}