Commit 34827943 by 曹润柘

合并分支 'master' 到 'caorunzhe'

Master

查看合并请求 !252
parents 9a971ca7 dd548e6b
...@@ -593,10 +593,10 @@ F(x)=\int_{-\infty}^x f(x)\textrm{d}x ...@@ -593,10 +593,10 @@ F(x)=\int_{-\infty}^x f(x)\textrm{d}x
% NEW SUBSUB-SECTION % NEW SUBSUB-SECTION
%---------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------
\subsubsection{2. 古德-图灵估计法} \subsubsection{2. 古德-图灵估计}
\vspace{-0.5em} \vspace{-0.5em}
\parinterval {\small\bfnew{古德-图灵估计}}\index{古德-图灵估计法}(Good-Turing Estimate)\index{Good-Turing Estimate}是Alan Turing和他的助手Irving John Good开发的,作为他们在二战期间破解德国密码机Enigma所使用方法的一部分,在1953 年Irving John Good将其发表。这一方法也是很多平滑算法的核心,其基本思路是:把非零的$n$元语法单元的概率降低,匀给一些低概率$n$元语法单元,以减小最大似然估计与真实概率之间的偏离\upcite{good1953population,gale1995good} \parinterval {\small\bfnew{古德-图灵估计}}\index{古德-图灵估计}(Good-Turing Estimate)\index{Good-Turing Estimate}是Alan Turing和他的助手Irving John Good开发的,作为他们在二战期间破解德国密码机Enigma所使用方法的一部分,在1953 年Irving John Good将其发表。这一方法也是很多平滑算法的核心,其基本思路是:把非零的$n$元语法单元的概率降低,匀给一些低概率$n$元语法单元,以减小最大似然估计与真实概率之间的偏离\upcite{good1953population,gale1995good}
\parinterval 假定在语料库中出现$r$次的$n$-gram有$n_r$个,特别的,出现0次的$n$-gram(即未登录词及词串)有$n_0$个。语料库中全部单词的总个数为$N$,显然: \parinterval 假定在语料库中出现$r$次的$n$-gram有$n_r$个,特别的,出现0次的$n$-gram(即未登录词及词串)有$n_0$个。语料库中全部单词的总个数为$N$,显然:
\begin{eqnarray} \begin{eqnarray}
......
...@@ -10,8 +10,8 @@ ...@@ -10,8 +10,8 @@
\node [anchor=west,inner sep=2pt] (eq1) at (0,0) {$f(s_u|t_v)$}; \node [anchor=west,inner sep=2pt] (eq1) at (0,0) {$f(s_u|t_v)$};
\node [anchor=west] (eq2) at (eq1.east) {$=$\ }; \node [anchor=west] (eq2) at (eq1.east) {$=$\ };
\draw [-] ([xshift=0.3em]eq2.east) -- ([xshift=11.6em]eq2.east); \draw [-] ([xshift=0.3em]eq2.east) -- ([xshift=11.6em]eq2.east);
\node [anchor=south west] (eq3) at ([xshift=1em]eq2.east) {$\sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})$}; \node [anchor=south west] (eq3) at ([xshift=1em]eq2.east) {$\sum_{i=1}^{K} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})$};
\node [anchor=north west] (eq4) at (eq2.east) {$\sum_{s_u} \sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})$}; \node [anchor=north west] (eq4) at (eq2.east) {$\sum_{s_u} \sum_{i=1}^{K} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})$};
{ {
\node [anchor=south] (label1) at ([yshift=-6em,xshift=3em]eq1.north west) {利用这个公式计算}; \node [anchor=south] (label1) at ([yshift=-6em,xshift=3em]eq1.north west) {利用这个公式计算};
......
...@@ -6,17 +6,17 @@ ...@@ -6,17 +6,17 @@
%------------------------------------------------------------------------- %-------------------------------------------------------------------------
\begin{tikzpicture} \begin{tikzpicture}
\node [anchor=north west] (line1) at (0,0) {\small\sffamily\bfseries{IBM模型1的训练(EM算法)}}; \node [anchor=north west] (line1) at (0,0) {\small\sffamily\bfseries{IBM模型1的训练(EM算法)}};
\node [anchor=north west] (line2) at ([yshift=-0.3em]line1.south west) {输入: 平行语料${(\seq{s}^{[1]},\seq{t}^{[1]}),...,(\seq{s}^{[N]},\seq{t}^{[N]})}$}; \node [anchor=north west] (line2) at ([yshift=-0.3em]line1.south west) {输入: 平行语料${(\seq{s}^{[1]},\seq{t}^{[1]}),...,(\seq{s}^{[K]},\seq{t}^{[K]})}$};
\node [anchor=north west] (line3) at ([yshift=-0.1em]line2.south west) {输出: 参数$f(\cdot|\cdot)$的最优值}; \node [anchor=north west] (line3) at ([yshift=-0.1em]line2.south west) {输出: 参数$f(\cdot|\cdot)$的最优值};
\node [anchor=north west] (line4) at ([yshift=-0.1em]line3.south west) {1: \textbf{Function} \textsc{EM}($\{(\seq{s}^{[1]},\seq{t}^{[1]}),...,(\seq{s}^{[N]},\seq{t}^{[N]})\}$) }; \node [anchor=north west] (line4) at ([yshift=-0.1em]line3.south west) {1: \textbf{Function} \textsc{EM}($\{(\seq{s}^{[1]},\seq{t}^{[1]}),...,(\seq{s}^{[K]},\seq{t}^{[K]})\}$) };
\node [anchor=north west] (line5) at ([yshift=-0.1em]line4.south west) {2: \ \ Initialize $f(\cdot|\cdot)$ \hspace{5em} $\rhd$ 比如给$f(\cdot|\cdot)$一个均匀分布}; \node [anchor=north west] (line5) at ([yshift=-0.1em]line4.south west) {2: \ \ Initialize $f(\cdot|\cdot)$ \hspace{5em} $\rhd$ 比如给$f(\cdot|\cdot)$一个均匀分布};
\node [anchor=north west] (line6) at ([yshift=-0.1em]line5.south west) {3: \ \ Loop until $f(\cdot|\cdot)$ converges}; \node [anchor=north west] (line6) at ([yshift=-0.1em]line5.south west) {3: \ \ Loop until $f(\cdot|\cdot)$ converges};
\node [anchor=north west] (line7) at ([yshift=-0.1em]line6.south west) {4: \ \ \ \ \textbf{foreach} $k = 1$ to $N$ \textbf{do}}; \node [anchor=north west] (line7) at ([yshift=-0.1em]line6.south west) {4: \ \ \ \ \textbf{foreach} $k = 1$ to $K$ \textbf{do}};
\node [anchor=north west] (line8) at ([yshift=-0.1em]line7.south west) {5: \ \ \ \ \ \ \ \footnotesize{$c_{\mathbb{E}}(\seq{s}_u|\seq{t}_v;\seq{s}^{[k]},\seq{t}^{[k]}) = \sum\limits_{j=1}^{|\seq{s}^{[k]}|} \delta(s_j,s_u) \sum\limits_{i=0}^{|\seq{t}^{[k]}|} \delta(t_i,t_v) \cdot \frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}\normalsize{}}; \node [anchor=north west] (line8) at ([yshift=-0.1em]line7.south west) {5: \ \ \ \ \ \ \ \footnotesize{$c_{\mathbb{E}}(\seq{s}_u|\seq{t}_v;\seq{s}^{[k]},\seq{t}^{[k]}) = \sum\limits_{j=1}^{|\seq{s}^{[k]}|} \delta(s_j,s_u) \sum\limits_{i=0}^{|\seq{t}^{[k]}|} \delta(t_i,t_v) \cdot \frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}\normalsize{}};
\node [anchor=north west] (line9) at ([yshift=-0.1em]line8.south west) {6: \ \ \ \ \textbf{foreach} $t_v$ appears at least one of $\{\seq{t}^{[1]},...,\seq{t}^{[N]}\}$ \textbf{do}}; \node [anchor=north west] (line9) at ([yshift=-0.1em]line8.south west) {6: \ \ \ \ \textbf{foreach} $t_v$ appears at least one of $\{\seq{t}^{[1]},...,\seq{t}^{[K]}\}$ \textbf{do}};
\node [anchor=north west] (line10) at ([yshift=-0.1em]line9.south west) {7: \ \ \ \ \ \ \ $\lambda_{t_v}^{'} = \sum_{s_u} \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;\seq{s}^{[k]},\seq{t}^{[k]})$}; \node [anchor=north west] (line10) at ([yshift=-0.1em]line9.south west) {7: \ \ \ \ \ \ \ $\lambda_{t_v}^{'} = \sum_{s_u} \sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\seq{s}^{[k]},\seq{t}^{[k]})$};
\node [anchor=north west] (line11) at ([yshift=-0.1em]line10.south west) {8: \ \ \ \ \ \ \ \textbf{foreach} $s_u$ appears at least one of $\{\seq{s}^{[1]},...,\seq{s}^{[N]}\}$ \textbf{do}}; \node [anchor=north west] (line11) at ([yshift=-0.1em]line10.south west) {8: \ \ \ \ \ \ \ \textbf{foreach} $s_u$ appears at least one of $\{\seq{s}^{[1]},...,\seq{s}^{[K]}\}$ \textbf{do}};
\node [anchor=north west] (line12) at ([yshift=-0.1em]line11.south west) {9: \ \ \ \ \ \ \ \ \ $f(s_u|t_v) = \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;\seq{s}^{[k]},\seq{t}^{[k]}) \cdot (\lambda_{t_v}^{'})^{-1}$}; \node [anchor=north west] (line12) at ([yshift=-0.1em]line11.south west) {9: \ \ \ \ \ \ \ \ \ $f(s_u|t_v) = \sum_{k=1}^{K} c_{\mathbb{E}}(s_u|t_v;\seq{s}^{[k]},\seq{t}^{[k]}) \cdot (\lambda_{t_v}^{'})^{-1}$};
\node [anchor=north west] (line13) at ([yshift=-0.1em]line12.south west) {10: \ \textbf{return} $f(\cdot|\cdot)$}; \node [anchor=north west] (line13) at ([yshift=-0.1em]line12.south west) {10: \ \textbf{return} $f(\cdot|\cdot)$};
\begin{pgfonlayer}{background} \begin{pgfonlayer}{background}
......
...@@ -50,7 +50,7 @@ IBM模型由Peter F. Brown等人于上世纪九十年代初提出\upcite{DBLP:jo ...@@ -50,7 +50,7 @@ IBM模型由Peter F. Brown等人于上世纪九十年代初提出\upcite{DBLP:jo
\end{figure} \end{figure}
%---------------------------------------------- %----------------------------------------------
\parinterval 上面的例子反映了人在做翻译时所使用的一些知识:首先,两种语言单词的顺序可能不一致,而且译文需要符合目标语的习惯,这也就是常说的翻译的{\small\sffamily\bfseries{流畅度}}\index{流畅度}问题(Fluency)\index{Fluency};其次,源语言单词需要准确地被翻译出来,也就是常说的翻译的{\small\sffamily\bfseries{准确性}}\index{准确性}(Accuracy)\index{Accuracy}问题和{\small\sffamily\bfseries{充分性}}\index{充分性}(Adequacy)\index{Adequacy}问题。为了达到以上目的,传统观点认为翻译过程需要包含三个步骤\upcite{parsing2009speech} \parinterval 上面的例子反映了人在做翻译时所使用的一些知识:首先,两种语言单词的顺序可能不一致,而且译文需要符合目标语的习惯,这也就是常说的翻译的{\small\sffamily\bfseries{流畅度}}\index{流畅度}问题(Fluency)\index{Fluency};其次,源语言单词需要准确地被翻译出来,也就是常说的翻译的{\small\sffamily\bfseries{准确性}}\index{准确性}(Accuracy)\index{Accuracy}问题和{\small\sffamily\bfseries{充分性}}\index{充分性}(Adequacy)\index{Adequacy}问题。为了达到以上目的,传统观点认为翻译过程需要包含三个步骤\upcite{parsing2009speech}
\begin{itemize} \begin{itemize}
\vspace{0.5em} \vspace{0.5em}
...@@ -273,13 +273,13 @@ $\seq{t}$ = machine\; \underline{translation}\; is\; a\; process\; of\; generati ...@@ -273,13 +273,13 @@ $\seq{t}$ = machine\; \underline{translation}\; is\; a\; process\; of\; generati
\subsubsection{3. 如何从大量的双语平行数据中进行学习?} \subsubsection{3. 如何从大量的双语平行数据中进行学习?}
\parinterval 如果有更多的句子,上面的方法同样适用。假设,有$N$个互译句对$\{(\seq{s}^{[1]},\seq{t}^{[1]})$,...,\\$(\seq{s}^{[N]},\seq{t}^{[N]})\}$。仍然可以使用基于相对频次的方法估计翻译概率$\funp{P}(x,y)$,具体方法如下: \parinterval 如果有更多的句子,上面的方法同样适用。假设,有$K$个互译句对$\{(\seq{s}^{[1]},\seq{t}^{[1]})$,...,\\$(\seq{s}^{[K]},\seq{t}^{[K]})\}$。仍然可以使用基于相对频次的方法估计翻译概率$\funp{P}(x,y)$,具体方法如下:
\begin{eqnarray} \begin{eqnarray}
\funp{P}(x,y) = \frac{{\sum_{i=1}^{N} c(x,y;\seq{s}^{[i]},\seq{t}^{[i]})}}{\sum_{i=1}^{N}{{\sum_{x',y'} c(x',y';\seq{s}^{[i]},\seq{t}^{[i]})}}} \funp{P}(x,y) = \frac{{\sum_{i=1}^{K} c(x,y;\seq{s}^{[i]},\seq{t}^{[i]})}}{\sum_{i=1}^{K}{{\sum_{x',y'} c(x',y';\seq{s}^{[i]},\seq{t}^{[i]})}}}
\label{eq:5-4} \label{eq:5-4}
\end{eqnarray} \end{eqnarray}
\parinterval 与公式\ref{eq:5-1}相比,公式\ref{eq:5-4}的分子、分母都多了一项累加符号$\sum_{i=1}^{N} \cdot$,它表示遍历语料库中所有的句对。换句话说,当计算词的共现次数时,需要对每个句对上的计数结果进行累加。从统计学习的角度,使用更大规模的数据进行参数估计可以提高结果的可靠性。计算单词的翻译概率也是一样,在小规模的数据上看,很多翻译现象的特征并不突出,但是当使用的数据量增加到一定程度,翻译的规律会很明显的体现出来。 \parinterval 与公式\ref{eq:5-1}相比,公式\ref{eq:5-4}的分子、分母都多了一项累加符号$\sum_{i=1}^{K} \cdot$,它表示遍历语料库中所有的句对。换句话说,当计算词的共现次数时,需要对每个句对上的计数结果进行累加。从统计学习的角度,使用更大规模的数据进行参数估计可以提高结果的可靠性。计算单词的翻译概率也是一样,在小规模的数据上看,很多翻译现象的特征并不突出,但是当使用的数据量增加到一定程度,翻译的规律会很明显的体现出来。
\parinterval 举个例子,实例\ref{eg:5-2}展示了一个由两个句对构成的平行语料库。 \parinterval 举个例子,实例\ref{eg:5-2}展示了一个由两个句对构成的平行语料库。
...@@ -633,7 +633,7 @@ g(\seq{s},\seq{t}) \equiv \prod_{j,i \in \widehat{A}}{\funp{P}(s_j,t_i)} \times ...@@ -633,7 +633,7 @@ g(\seq{s},\seq{t}) \equiv \prod_{j,i \in \widehat{A}}{\funp{P}(s_j,t_i)} \times
\end{figure} \end{figure}
%---------------------------------------------- %----------------------------------------------
\item 源语言单词可以翻译为空,这时它对应到一个虚拟或伪造的目标语单词$t_0$。在图\ref{fig:5-16}所示的例子中,``在''没有对应到``on the table''中的任意一个词,而是把它对应到$t_0$上。这样,所有的源语言单词都能找到一个目标语单词对应。这种设计也很好地引入了{\small\sffamily\bfseries{空对齐}}\index{空对齐}的思想,即源语言单词不对应任何真实存在的单词的情况。而这种空对齐的情况在翻译中是频繁出现的,比如虚词的翻译。 \item 源语言单词可以翻译为空,这时它对应到一个虚拟或伪造的目标语单词$t_0$。在图\ref{fig:5-16}所示的例子中,``在''没有对应到``on the table''中的任意一个词,而是把它对应到$t_0$上。这样,所有的源语言单词都能找到一个目标语单词对应。这种设计也很好地引入了{\small\sffamily\bfseries{空对齐}}\index{空对齐}(Empty Alignment\index{Empty Alignment}的思想,即源语言单词不对应任何真实存在的单词的情况。而这种空对齐的情况在翻译中是频繁出现的,比如虚词的翻译。
%---------------------------------------------- %----------------------------------------------
\begin{figure}[htp] \begin{figure}[htp]
...@@ -703,7 +703,7 @@ g(\seq{s},\seq{t}) \equiv \prod_{j,i \in \widehat{A}}{\funp{P}(s_j,t_i)} \times ...@@ -703,7 +703,7 @@ g(\seq{s},\seq{t}) \equiv \prod_{j,i \in \widehat{A}}{\funp{P}(s_j,t_i)} \times
\subsection{基于词对齐的翻译实例} \subsection{基于词对齐的翻译实例}
\parinterval 用前面图\ref{fig:5-16}中例子来对公式\ref{eq:5-18}进行说明。例子中,源语言句子``在\ \ 桌子\ \ 上''目标语译文``on the table''之间的词对齐为$\seq{a}=\{\textrm{1-0, 2-3, 3-1}\}$。公式\ref{eq:5-18}的计算过程如下: \parinterval 用前面图\ref{fig:5-16}中例子来对公式\ref{eq:5-18}进行说明。例子中,源语言句子``在\ \ 桌子\ \ 上''目标语译文``on the table''之间的词对齐为$\seq{a}=\{\textrm{1-0, 2-3, 3-1}\}$ 公式\ref{eq:5-18}的计算过程如下:
\begin{itemize} \begin{itemize}
\vspace{0.5em} \vspace{0.5em}
...@@ -720,11 +720,11 @@ g(\seq{s},\seq{t}) \equiv \prod_{j,i \in \widehat{A}}{\funp{P}(s_j,t_i)} \times ...@@ -720,11 +720,11 @@ g(\seq{s},\seq{t}) \equiv \prod_{j,i \in \widehat{A}}{\funp{P}(s_j,t_i)} \times
\funp{P}(\seq{s},\seq{a}|\seq{t})\; &= & \funp{P}(m|\seq{t}) \prod\limits_{j=1}^{m} \funp{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,\seq{t}) \funp{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,\seq{t}) \nonumber \\ \funp{P}(\seq{s},\seq{a}|\seq{t})\; &= & \funp{P}(m|\seq{t}) \prod\limits_{j=1}^{m} \funp{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,\seq{t}) \funp{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,\seq{t}) \nonumber \\
&=&\funp{P}(m=3 \mid \textrm{$t_0$ on the table}){\times} \nonumber \\ &=&\funp{P}(m=3 \mid \textrm{$t_0$ on the table}){\times} \nonumber \\
&&{\funp{P}(a_1=0 \mid \phi,\phi,3,\textrm{$t_0$ on the table}){\times} } \nonumber \\ &&{\funp{P}(a_1=0 \mid \phi,\phi,3,\textrm{$t_0$ on the table}){\times} } \nonumber \\
&&{\funp{P}(f_1=\textrm{} \mid \textrm{\{1-0\}},\phi,3,\textrm{$t_0$ on the table}){\times} } \nonumber \\ &&{\funp{P}(s_1=\textrm{} \mid \textrm{\{1-0\}},\phi,3,\textrm{$t_0$ on the table}){\times} } \nonumber \\
&&{\funp{P}(a_2=3 \mid \textrm{\{1-0\}},\textrm{},3,\textrm{$t_0$ on the table}) {\times}} \nonumber \\ &&{\funp{P}(a_2=3 \mid \textrm{\{1-0\}},\textrm{},3,\textrm{$t_0$ on the table}) {\times}} \nonumber \\
&&{\funp{P}(f_2=\textrm{桌子} \mid \textrm{\{1-0, 2-3\}},\textrm{},3,\textrm{$t_0$ on the table}) {\times}} \nonumber \\ &&{\funp{P}(s_2=\textrm{桌子} \mid \textrm{\{1-0, 2-3\}},\textrm{},3,\textrm{$t_0$ on the table}) {\times}} \nonumber \\
&&{\funp{P}(a_3=1 \mid \textrm{\{1-0, 2-3\}},\textrm{\ \ 桌子},3,\textrm{$t_0$ on the table}) {\times}} \nonumber \\ &&{\funp{P}(a_3=1 \mid \textrm{\{1-0, 2-3\}},\textrm{\ \ 桌子},3,\textrm{$t_0$ on the table}) {\times}} \nonumber \\
&&{\funp{P}(f_3=\textrm{} \mid \textrm{\{1-0, 2-3, 3-1\}},\textrm{\ \ 桌子},3,\textrm{$t_0$ on the table}) } &&{\funp{P}(s_3=\textrm{} \mid \textrm{\{1-0, 2-3, 3-1\}},\textrm{\ \ 桌子},3,\textrm{$t_0$ on the table}) }
\label{eq:5-19} \label{eq:5-19}
\end{eqnarray} \end{eqnarray}
...@@ -1064,10 +1064,10 @@ f(s_u|t_v)=\frac{c_{\mathbb{E}}(s_u|t_v;\seq{s},\seq{t})} { \sum\limits_{s_u} c ...@@ -1064,10 +1064,10 @@ f(s_u|t_v)=\frac{c_{\mathbb{E}}(s_u|t_v;\seq{s},\seq{t})} { \sum\limits_{s_u} c
\end{figure} \end{figure}
%---------------------------------------------- %----------------------------------------------
\parinterval 进一步,假设有$N$个互译的句对(称作平行语料): \parinterval 进一步,假设有$K$个互译的句对(称作平行语料):
$\{(\seq{s}^{[1]},\seq{t}^{[1]}),...,(\seq{s}^{[N]},\seq{t}^{[N]})\}$$f(s_u|t_v)$的期望频次为: $\{(\seq{s}^{[1]},\seq{t}^{[1]}),...,(\seq{s}^{[K]},\seq{t}^{[K]})\}$$f(s_u|t_v)$的期望频次为:
\begin{eqnarray} \begin{eqnarray}
c_{\mathbb{E}}(s_u|t_v)=\sum\limits_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]}) c_{\mathbb{E}}(s_u|t_v)=\sum\limits_{i=1}^{K} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})
\label{eq:5-46} \label{eq:5-46}
\end{eqnarray} \end{eqnarray}
......
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