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\title{\Large{��"��"��ʼ�����ڵ��ʵķ���ģ��}}
\author{\large{\textbf{Фͩ\ \ �쾸��}}}
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\blue{\url{xiaotong@mail.neu.edu.cn}} \black{} \\
\blue{\url{zhujingbo@mail.neu.edu.cn}} \black{} \\
\vspace{1.0em}
������ѧ ��Ȼ���Դ���ʵ���� \\
\blue{\underline{\url{http://www.nlplab.com}}} \black{} \\
\vspace{0.2cm}
\hspace{0.1cm} \includegraphics[scale=0.1]{../Figures/logo.pdf}
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\date{}

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%%%------------------------------------------------------------------------------------------------------------
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%%%------------------------------------------------------------------------------------------------------------
\section{�����ʷ��룺һ��ʵ��}

%%%------------------------------------------------------------------------------------------------------------
\subsection{����˼��}

%%%------------------------------------------------------------------------------------------------------------
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�����������֮�����һ��\alert{����}��Ķ�Ӧ��������ķ�������\alert{����}�ķ���"���"
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\item&lt;3-&gt; ��ͳ�۵��µ�(���ڴ�)�ķ������ - ��������\\
\vspace{-0.8em}
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\item \textbf{����}����������ӽ��зִ�
\vspace{-0.5em}
\item&lt;4-&gt; \textbf{ת��}�����ÿ��Դ�ﵥ�ʵķ���
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\item&lt;5-&gt; \textbf{����}�������ʵķ������ͨ˳����������
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%%%------------------------------------------------------------------------------------------------------------
\subsection{�������̣��˹� vs. ����}

%%%------------------------------------------------------------------------------------------------------------
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%%%------------------------------------------------------------------------------------------------------------
%%% ����ʾ�� - ��������
\begin{frame}{��������ϵͳ����ν��з���ģ�}

\vspace{-0.5em}


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\begin{scope}

\invisible{
\node [anchor=north west] (label1) at (ft11.south west) {\small{ѡ����ѵ����ģ����������򣬵õ������Ľ��}};
}

\visible&lt;8-&gt;{
\draw[decorate,thick,decoration={brace,amplitude=5pt,mirror}] ([yshift=8em,xshift=-0.5em]t13.south west) -- ([xshift=-0.5em]t13.south west) node [pos=0.5,left,xshift=-0.5em,yshift=2.0em] (label2) {\footnotesize{\textbf{��˫����}}};
\node [anchor=north west] (label2part2) at ([yshift=0.3em]label2.south west) {\footnotesize{\textbf{�����Զ�}}};
\node [anchor=north west] (label2part3) at ([yshift=0.3em]label2part2.south west) {\footnotesize{\textbf{ѧϰ�ʵ�}}};
\node [anchor=north west] (label2part4) at ([yshift=0.3em]label2part3.south west) {\footnotesize{\textbf{(ѵ��)}}};
}

\visible&lt;8-&gt;{
\draw[decorate,thick,decoration={brace,amplitude=5pt,mirror}] ([yshift=-0.2em,xshift=-0.5em]t13.south west) -- ([yshift=-9.5em,xshift=-0.5em]t13.south west) node [pos=0.5,left,xshift=-0.5em,yshift=2.0em] (label3) {\footnotesize{\textbf{���ø���}}};
\node [anchor=north west] (label3part2) at ([yshift=0.3em]label3.south west) {\footnotesize{\textbf{���Ĵʵ�}}};
\node [anchor=north west] (label3part3) at ([yshift=0.3em]label3part2.south west) {\footnotesize{\textbf{���з���}}};
\node [anchor=north west] (label3part4) at ([yshift=0.3em]label3part3.south west) {\footnotesize{\textbf{(����)}}};
}

\visible&lt;1-7&gt;{
\node [anchor=east] (label4) at ([yshift=0.4em]ft11.west) {\scriptsize{�������һ��}};
\node [anchor=north west] (label4part2) at ([yshift=0.7em]label4.south west) {\scriptsize{����ѡ��·��}};
}

\visible&lt;2-7&gt;{
\node [anchor=east] (label5) at ([yshift=0.4em]ft21.west) {\scriptsize{��ͬ�����Ķ�}};
\node [anchor=north west] (label5part2) at ([yshift=0.7em]label5.south west) {\scriptsize{Ӧ��ͬ��·��}};
}

\visible&lt;3-7&gt;{
\node [anchor=east] (label6) at ([yshift=0.4em]ft31.west) {\scriptsize{���ʷ���Ĵ�}};
\node [anchor=north west] (label6part2) at ([yshift=0.7em]label6.south west) {\scriptsize{��Ҳ���ܲ�ͬ}};
}

\visible&lt;4-7&gt;{
\node [anchor=east] (label7) at ([yshift=0.4em]ft41.west) {\scriptsize{���ܵķ���·}};
\node [anchor=north west] (label7part2) at ([yshift=0.7em]label7.south west) {\scriptsize{���ǳ���}};
}

\end{scope}

\begin{scope}

\visible&lt;6-&gt;{
\node [anchor=west] (score1) at ([xshift=1.5em]ft14.east) {\footnotesize{P=0.042}};
\node [anchor=west] (score2) at ([xshift=1.5em]ft24.east) {\footnotesize{P=0.006}};
\node [anchor=west] (score3) at ([xshift=1.5em]ft34.east) {\footnotesize{P=0.003}};
\node [anchor=south] (scorelabel) at (score1.north) {\scriptsize{\textbf{\alert{������һ��ģ�͵÷�}}}};
\node [anchor=south] (scorelabel2) at ([yshift=-0.5em]scorelabel.north) {\scriptsize{\textbf{\alert{ϵͳ��ÿ������}}}};
}
\visible&lt;7-&gt;{
\node [anchor=north] (scorelabel2) at (score3.south) {\scriptsize{\textbf{ѡ��÷�}}};
\node [anchor=north west] (scorelabel2part2) at ([xshift=-0.5em,yshift=0.5em]scorelabel2.south west) {\scriptsize{\textbf{��ߵ�����}}};
\node [anchor=center,draw=ublue,circle,thick,fill=white,inner sep=1pt,circular drop shadow={shadow xshift=0.05em,shadow yshift=-0.05em}] (head1) at ([xshift=0.3em]score1.east) {\scriptsize{{\color{ugreen} \textbf{ok}}}};
}

\end{scope}

\begin{scope}

\visible&lt;2-&gt;{
\draw [-&gt;,ultra thick,ublue,line width=2pt,opacity=0.7] ([xshift=-0.5em]t13.west) -- ([xshift=0.8em]t13.east) -- ([xshift=-0.2em]t21.west) -- ([xshift=0.8em]t21.east) -- ([xshift=-0.2em]t31.west) -- ([xshift=0.8em]t31.east) -- ([xshift=-0.2em]t41.west) -- ([xshift=0.8em]t41.east) -- ([xshift=-0.2em]t51.west) -- ([xshift=1.2em]t51.east);
}

\draw [-&gt;,ultra thick,red,line width=2pt,opacity=0.7] ([xshift=-0.5em]t13.west) -- ([xshift=0.8em]t13.east) -- ([xshift=-0.2em]t22.west) -- ([xshift=0.8em]t22.east) -- ([xshift=-0.2em]t31.west) -- ([xshift=0.8em]t31.east) -- ([xshift=-0.2em]t41.west) -- ([xshift=0.8em]t41.east) -- ([xshift=-0.2em]t52.west) -- ([xshift=1.2em]t52.east);

\end{scope}

\end{tikzpicture}
\end{flushright}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{ʵ��������һ�����׻�������ϵͳ}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 1 learning word translation probablities
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ��ѧϰ���ʷ������}
\label{wordtranslationprob}

\begin{itemize}
\item ����һ���򵥵Ļ�������ϵͳ
    \begin{enumerate}
    \item ģ��ѧϰ - ��˫��ƽ��������ѧϰ���ʷ���֪ʶ
    \item&lt;2-&gt; ���� - ���õ��ʷ���֪ʶ�����µľ��ӽ��з���
    \end{enumerate}

\begin{tikzpicture}

{\scriptsize
\node [anchor=north west,inner sep=1pt] (entry1) at (0,0) {\tiny{\textbf{1:} �� �� ���� $\leftrightarrow$ This is data}};
\node [anchor=north west,inner sep=1pt] (entry2) at ([yshift=0.1em]entry1.south west) {\tiny{\textbf{2:} С�� ��$\leftrightarrow$ Look out !}};
\node [anchor=north west,inner sep=1pt] (entry3) at ([yshift=0.1em]entry2.south west) {\tiny{\textbf{3:} �� �� ˭ $\leftrightarrow$ Who are you}};
\node [anchor=north west,inner sep=2pt] (entry4) at ([yshift=0.1em]entry3.south west) {...};
\node [anchor=south west] (corpuslabel) at (entry1.north west) {{\color{ublue} \textbf{˫��ƽ������}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (entry1) (entry2) (entry3) (entry4) (corpuslabel)] (corpus) {};
\end{pgfonlayer}
}

\node [anchor=west,ugreen] (P) at ([xshift=4em,yshift=-0.7em]corpus.east){P($t|s$)};
\node [anchor=south] (modellabel) at (P.north) {{\color{ublue} {\scriptsize \textbf{����ģ��}}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (P) (modellabel)] (model) {};
\end{pgfonlayer}

\draw [-&gt;,very thick,ublue] ([xshift=0.2em]corpus.east) -- ([xshift=3.2em]corpus.east)  node [pos=0.5, above] {\alert{\scriptsize{ģ��ѧϰ}}};

\visible&lt;2-&gt;{
\draw [-&gt;,very thick,ublue] ([xshift=0.4em]model.east) -- ([xshift=3.4em]model.east)  node [inner sep=0pt,pos=0.5, above,yshift=0.3em] (decodingarrow) {\alert{\scriptsize{���\&amp;����}}};

{\scriptsize
\node [anchor=north west,inner sep=2pt] (sentlabel) at ([xshift=5.5em,yshift=-0.3em]model.north east) {{\color{ublue} \textbf{������������}}};
\node [anchor=north west] (sent) at ([yshift=-0.5em]sentlabel.south west) {\textbf{���������}};
\node [anchor=north west] (sentpart2) at ([yshift=0.3em]sent.south west) {\textbf{���з���}};
}
}

\begin{pgfonlayer}{background}
\visible&lt;2-&gt;{
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (sentlabel) (sent) (sentpart2)] (segsystem) {};
}
\end{pgfonlayer}


\visible&lt;3-&gt;{
\node[rectangle,fill=white,fill opacity=0.85,inner sep=0pt] [fit = (segsystem) (decodingarrow)] (segsystem2) {};
}

\end{tikzpicture}

\item&lt;3-&gt; \textbf{����1}���������ʷ���� - ����ʵ�\\
          \small{���������Դ���Ե���$x$��Ҫ��������п��ܵ�����$Y$������һ��������$(s,t)$������$y \in Y$������$\textrm{P}(x \leftrightarrow y; s, t)$��ʾ$x$��$y$��$(x,y)$�л���ĸ��ʣ�������$x$��$y$�����ϸ��ʱ�ʾ��

          \vspace{-2.0em}

          \begin{eqnarray}
          \textrm{P}(x \leftrightarrow y; s,t) &amp; \equiv &amp; \textrm{P}(x,y;s,t) \nonumber \\
                                          &amp; = &amp; \frac{c(x,y;s,t)}{\sum_{x',y'} c(x',y';s,t)} \nonumber
          \end{eqnarray}

          \vspace{-0.5em}

          $c(x,y;s,t)$��ʾ$(x,y)$��$(s,t)$�й��ֵĴ���; $\sum_{x',y'} c(x',y';s,t)$��ʾ$(s,t)$������Դ/���ĵ��ʹ��ֵ��ܴ���
          }

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 1 learning word translation probablities - for single sentence
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ��ѧϰ���ʷ������(2)}
\vspace{-1em}
\begin{eqnarray}
\textrm{P}(x,y;s,t) &amp; = &amp; \frac{c(x,y;s,t)}{\sum_{x',y'} c(x',y';s,t)} \nonumber
\end{eqnarray}

\vspace{-0.5em}

\begin{flushleft}
\begin{tikzpicture}

\node [anchor=west] (s) at (0,0) {$s=$};
\node [anchor=center] (sw1) at ([xshift=1em]s.east) {����};
\visible&lt;1,4-&gt;{
\node [anchor=center] (sw2) at ([xshift=1.3em]sw1.east) {����};
}
\node [anchor=center] (sw3) at ([xshift=5.5em]sw2.east) {�� �� �� ����� �� ����};
\visible&lt;1-3,6-&gt;{
\node [anchor=center] (sw4) at ([xshift=1.0em]sw3.east) {����};
}

\node [anchor=north west] (t) at (s.south west) {$t=$};
\node [anchor=center] (tw1) at ([xshift=1.8em]t.east) {machine};
\visible&lt;1,3,5-&gt;{
\node [anchor=center] (tw2) at ([xshift=2.2em]tw1.east) {translation};
}
\node [anchor=center] (tw3) at ([xshift=1.4em,yshift=-0.1em]tw2.east) {is just};
\visible&lt;1,2,4,6-&gt;{
\node [anchor=center] (tw4) at ([xshift=2.3em,yshift=0.1em]tw3.east) {translation};
}
\node [anchor=center] (tw5) at ([xshift=2.6em,yshift=-0.1em]tw4.east) {by computer};

\visible&lt;2-3&gt;{
\node [anchor=center,red] (sw2highlight) at ([xshift=1.3em]sw1.east) {\underline{����}};
}
\visible&lt;4-5&gt;{
\node [anchor=center,red] (sw4highlight) at ([xshift=1.0em]sw3.east) {\underline{����}};
}
\visible&lt;2,4&gt;{
\node [anchor=center,red] (tw2highlight) at ([xshift=2.2em]tw1.east) {\underline{translation}};
}
\visible&lt;3,5&gt;{
\node [anchor=center,red] (tw4highlight) at ([xshift=2.3em,yshift=0.1em]tw3.east) {\underline{translation}};
}

\end{tikzpicture}
\end{flushleft}

\vspace{-1.0em}

\begin{itemize}

\item $c(\textrm{'����'},\textrm{'translation'};s,t)=\only&lt;1&gt;{?}\visible&lt;2-&gt;{1}\visible&lt;3-&gt;{+1}\visible&lt;4-&gt;{+1}\visible&lt;5-&gt;{+1=4}$
\item&lt;6-&gt; $\sum_{x',y'} c(x',y';s,t)= \textrm{ʹ����...}  = 63\visible&lt;7-&gt;{ = 9 \times 7 = |s| \times |t|}$
    \vspace{0.3em}
    \begin{itemize}
    \item&lt;7-&gt; $|\cdot|$��ʾ���ӳ���
    \end{itemize}
    \vspace{0.3em}
\item&lt;8-&gt; '����'��'translation'�Ļ������Ϊ
\begin{displaymath}
\textrm{P}(\textrm{'����'},\textrm{'translation'};s,t) = 4/63
\end{displaymath}
\vspace{-0.5em}
���Ƶ�
\vspace{-0.5em}
\begin{eqnarray}
\textrm{P}(\textrm{'����'},\textrm{'translation'};s,t) &amp; = &amp; 2/63 \nonumber \\
\textrm{P}(\textrm{'����'},\textrm{'look'};s,t) &amp; = &amp; 0/63 \nonumber
\end{eqnarray}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 1 learning word translation probablities - for several sentences
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ��ѧϰ���ʷ������(3)}

\begin{itemize}

\item �ܶ�ʱ�������ж��������$(s^{[1]},t^{[1]}),...,(s^{[n]},t^{[n]})$����֮Ϊ\alert{˫��ƽ������(����)}��������ʿ��Ա�����Ϊ

\vspace{-1em}

\begin{eqnarray}
\textrm{P}(x,y) &amp; = &amp; \frac{\sum_{i=1}^{n}c(x,y;s^{[i]},t^{[i]})}{\sum_{i=1}^{n} \sum_{x',y'} c(x',y';s^{[i]},t^{[i]})} \nonumber
\end{eqnarray}

\item&lt;2-&gt; ˵���˾��Ǽ���$(x,y)$��Ƶ��ʱ����ÿ���������ۼ�

\begin{flushleft}
\begin{tikzpicture}

\node [anchor=west] (s1) at (0,0) {$s_1=$ ���� ���� �� �� �� ����� ���� ����};
\node [anchor=north west] (t1) at ([yshift=0.4em]s1.south west) {$t_1=$ Machine translation is just translation by computer};

\node [anchor=north west] (s2) at (t1.south west) {$s_2=$ �� �˹� ���� �� ?};
\node [anchor=north west] (t2) at ([yshift=0.4em]s2.south west) {$t_2=$ So , what is human translation ?};

\end{tikzpicture}
\end{flushleft}

\vspace{-1.5em}

{\footnotesize
\begin{eqnarray}
&amp;   &amp; \textrm{P}(\textrm{'����'},\textrm{'translation'}) \nonumber \\
&amp; = &amp; \frac{c(\textrm{'����'},\textrm{'translation'};s^{[1]},t^{[1]})+c(\textrm{'����'},\textrm{'translation'};s^{[2]},t^{[2]})}{\sum_{x',y'} c(x',y';s^{[1]},t^{[1]}) + \sum_{x',y'} c(x',y';s^{[2]},t^{[2]})} \nonumber \\
\visible&lt;3-&gt;{&amp; = &amp; \frac{4 + 1}{|s^{[1]}| \times |t^{[1]}| + |s^{[2]}| \times |t^{[2]}|} = \frac{4 + 1}{9 \times 7 + 5 \times 7} = \frac{5}{98}} \nonumber
\end{eqnarray}
}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 2 modeling sentence translaiton probability
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ�����Ӽ�����ģ��}

\begin{center}
\begin{tikzpicture}

{\scriptsize
\node [anchor=north west,inner sep=1pt] (entry1) at (0,0) {\tiny{\textbf{1:} �� �� ���� $\leftrightarrow$ This is data}};
\node [anchor=north west,inner sep=1pt] (entry2) at ([yshift=0.1em]entry1.south west) {\tiny{\textbf{2:} С�� ��$\leftrightarrow$ Look out !}};
\node [anchor=north west,inner sep=1pt] (entry3) at ([yshift=0.1em]entry2.south west) {\tiny{\textbf{3:} �� �� ˭ $\leftrightarrow$ Who are you}};
\node [anchor=north west,inner sep=2pt] (entry4) at ([yshift=0.1em]entry3.south west) {...};
\node [anchor=south west] (corpuslabel) at (entry1.north west) {{\color{ublue} \textbf{˫��ƽ������}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (entry1) (entry2) (entry3) (entry4) (corpuslabel)] (corpus) {};
\end{pgfonlayer}
}

\node [anchor=west,ugreen] (P) at ([xshift=4em,yshift=-0.7em]corpus.east){P($t|s$)};
\node [anchor=south] (modellabel) at (P.north) {{\color{ublue} {\scriptsize \textbf{����ģ��}}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (P) (modellabel)] (model) {};
\end{pgfonlayer}

\draw [-&gt;,very thick,ublue] ([xshift=0.2em]corpus.east) -- ([xshift=3.0em]corpus.east)  node [inner sep=0pt,pos=0.5,above,yshift=0.3em] (trainingarrow) {\alert{\scriptsize{ģ��ѧϰ}}};

\draw [-&gt;,very thick,ublue] ([xshift=0.4em]model.east) -- ([xshift=3.4em]model.east)  node [inner sep=0pt,pos=0.5,above,yshift=0.3em] (decodingarrow) {\alert{\scriptsize{���\&amp;����}}};

{\scriptsize
\node [anchor=north west,inner sep=2pt] (sentlabel) at ([xshift=5.5em,yshift=-0.3em]model.north east) {{\color{ublue} \textbf{������������}}};
\node [anchor=north west] (sent) at ([yshift=-0.5em]sentlabel.south west) {\textbf{���������}};
\node [anchor=north west] (sentpart2) at ([yshift=0.3em]sent.south west) {\textbf{���з���}};
}

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (sentlabel) (sent) (sentpart2)] (segsystem) {};
\end{pgfonlayer}

\node[rectangle,fill=white,fill opacity=0.85,inner sep=0pt] [fit = (segsystem) (decodingarrow)] (segsystem2) {};
\node[rectangle,fill=white,fill opacity=0.85,inner sep=0pt] [fit = (corpus) (trainingarrow)] (corpus2) {};

\end{tikzpicture}
\end{center}

\begin{itemize}
\item \textbf{����2}: ������ľ��$(s,t)$������Ӽ��������$\textrm{P}(t|s)$ \\

      \vspace{0.5em}
      \visible&lt;2-&gt;{
      ��һ�ֱȽϼ򵥵�˼·������$(s,t)$�ϵ�һ�ַ���$g(s,t)$
      \begin{itemize}
      \item $g(s,t)$��ֵԽ��������Խ��
      \item $g(s,t)$��ֵԽС��������Խ��
      \end{itemize}
      }

      \vspace{0.5em}

      \visible&lt;3-&gt;{
      ���ǣ����ǽ�һ������

      \begin{displaymath}
      \textrm{P}(t|s) = \frac{g(s,t)}{\sum_{t'}g(s,t')}
      \end{displaymath}

      ʵ���Ͼ��Ƕ�$g(s,t)$�����п��ܵ����ļ���������һ����ʹ����и�������
      }

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 2 modeling sentence translaiton probability - word alignment
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ�����Ӽ�����ģ��(2)}

\vspace{0.5em}

\begin{spacing}{1.1}
\begin{itemize}

\item ��������
    \begin{enumerate}
    \item ��μ���$g(s,t)$ \visible&lt;2-&gt;{- ��ؼ��Ľ�ģ���⣬���Ͽ�ʼ}
    \item ��μ���$\sum_{t'} g(s,t')$ \visible&lt;2-&gt;{- ʵ����\alert{���ü���}��������˵}
    \end{enumerate}

\item&lt;3-&gt; \textbf{��$g(s,t)$��ģ: }���ݱ��µ�һҳ�ļ��裬$s$��$t$֮�����һ�ֵ��ʼ�Ķ�Ӧ�����ǳ�֮Ϊ\alert{�ʶ���}��ϵ

\begin{center}
\begin{tikzpicture}

\begin{scope}
\node [anchor=west] (s1) at (0,0) {��\footnotesize{$_1$}};
\node [anchor=west] (s2) at ([xshift=0.5em]s1.east) {��\footnotesize{$_2$}};
\node [anchor=west] (s3) at ([xshift=0.5em]s2.east) {��\footnotesize{$_3$}};
\node [anchor=west] (s4) at ([xshift=0.5em]s3.east) {�е�\footnotesize{$_4$}};
\node [anchor=west] (s5) at ([xshift=0.5em]s4.east) {����\footnotesize{$_5$}};
\node [anchor=east] (s) at (s1.west) {$s=$};
\end{scope}

\begin{scope}[yshift=-3.0em]
\node [anchor=west] (t1) at (0.35em,0) {I\footnotesize{$_1$}};
\node [anchor=west] (t2) at ([xshift=0.3em,yshift=-0.1em]t1.east) {am\footnotesize{$_2$}};
\node [anchor=west] (t3) at ([xshift=0.3em,yshift=0.1em]t2.east) {satisfied\footnotesize{$_3$}};
\node [anchor=west] (t4) at ([xshift=0.3em]t3.east) {with\footnotesize{$_4$}};
\node [anchor=west] (t5) at ([xshift=0.3em,yshift=-0.2em]t4.east) {you\footnotesize{$_5$}};
\node [anchor=east] (t) at ([xshift=-0.3em]t1.west) {$t=$};
\end{scope}


\draw [-,thick,ublue,dashed] (s1.south) -- (t1.north);
\draw [-,thick,ublue,dashed] (s4.south) -- ([yshift=0.3em]t2.north);
\draw [-,thick,ublue,dashed] (s2.south) ..controls +(south:1em) and +(north:1em).. (t4.north);
\draw [-,thick,ublue,dashed] (s3.south) ..controls +(south:0.5em) and +(north:1.5em).. (t5.north);
\draw [-,thick,ublue,dashed] (s5.south) -- (t3.north);

\end{tikzpicture}
\end{center}

ÿ�����ߴ���һ��\alert{��������}����Ϊ$(j,i)$����˼��Դ���Եĵ�$j$�����ʶ�ӦĿ�����Ե�$i$�����ʣ���$s_j$��$t_i$��Ӧ��

\item&lt;3-&gt; ���ж������ӹ��ɼ���$A$������������
      $A=\{(1,1),(2,4),(3,5),(4,2),(5,3)\}$

\end{itemize}
\end{spacing}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 2 modeling sentence translaiton probability - model score
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ�����Ӽ�����ģ��(3)}

\begin{itemize}
\item ����һ�����$(s,t)$��������֮���(����)�ʶ���$\hat{A}$�����Զ���ģ�͵÷�Ϊ��
\begin{displaymath}
g(s,t) \equiv \prod_{(j,i) \in \hat{A}} \textrm{P}(s_j,t_i)
\end{displaymath}
      ��Ȼÿ�����ʷ�����ʶ��ߣ���ô�����ģ�͵÷�Ҳ��

\begin{center}
\begin{tikzpicture}

\begin{scope}
\node [anchor=west] (s1) at (0,0) {��\footnotesize{$_1$}};
\node [anchor=west] (s2) at ([xshift=0.5em]s1.east) {��\footnotesize{$_2$}};
\node [anchor=west] (s3) at ([xshift=0.5em]s2.east) {��\footnotesize{$_3$}};
\node [anchor=west] (s4) at ([xshift=0.5em]s3.east) {�е�\footnotesize{$_4$}};
\node [anchor=west] (s5) at ([xshift=0.5em]s4.east) {����\footnotesize{$_5$}};
\node [anchor=east] (s) at (s1.west) {$s=$};
\end{scope}

\begin{scope}[yshift=-3.0em]
\node [anchor=west] (t1) at (0.35em,0) {I\footnotesize{$_1$}};
\node [anchor=west] (t2) at ([xshift=0.3em,yshift=-0.1em]t1.east) {am\footnotesize{$_2$}};
\node [anchor=west] (t3) at ([xshift=0.3em,yshift=0.1em]t2.east) {satisfied\footnotesize{$_3$}};
\node [anchor=west] (t4) at ([xshift=0.3em]t3.east) {with\footnotesize{$_4$}};
\node [anchor=west] (t5) at ([xshift=0.3em,yshift=-0.2em]t4.east) {you\footnotesize{$_5$}};
\node [anchor=east] (t) at ([xshift=-0.3em]t1.west) {$t=$};
\end{scope}


\draw [-,thick,ublue,dashed] (s1.south) -- (t1.north);
\draw [-,thick,ublue,dashed] (s4.south) -- ([yshift=0.3em]t2.north);
\draw [-,thick,ublue,dashed] (s2.south) ..controls +(south:1em) and +(north:1em).. (t4.north);
\draw [-,thick,ublue,dashed] (s3.south) ..controls +(south:0.5em) and +(north:1.5em).. (t5.north);
\draw [-,thick,ublue,dashed] (s5.south) -- (t3.north);

\end{tikzpicture}
\end{center}

\vspace{-2.5em}

\begin{eqnarray}
g(s,t) &amp; = &amp; \textrm{P}(\textrm{'��','I'}) \times \textrm{P}(\textrm{'��','with'}) \times \textrm{P}(\textrm{'��','you'}) \times \nonumber \\
       &amp;   &amp; \textrm{P}(\textrm{'�е�','am'}) \times \textrm{P}(\textrm{'����','satisfied'}) \nonumber
\end{eqnarray}

\item&lt;2-&gt; \textbf{��������}������е��ʷ���Ϊ����ô�죿
    \begin{itemize}
    \item �򵥵Ľ��������ƽ�����������ַ���һ����С�ĸ���ֵ
    \item �����ڽ�ģ�п��ǣ��������IBMģ��
    \end{itemize}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 2 modeling sentence translaiton probability - introducing LM into the model
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ�����Ӽ�����ģ��(4)}

\begin{itemize}
\item \textbf{����}��������Ƶ�$g(s,t)$û�п��Ǵ������Ϣ����ͬ��ʳ����ڲ�ͬ��λ�ã��÷���ͬ - �޷�ѡ������������

\vspace{0.5em}

\begin{tabular}{l | l }

&amp; \footnotesize{$\prod\limits_{(j,i) \in \hat{A}} \textrm{P}(s_j,t_i)$} \visible&lt;2-&gt;{\alert{\footnotesize{$\times\textrm{P}_{lm}(t)$}}} \\ \hline

\begin{tikzpicture}

\begin{scope}

{\footnotesize
\begin{scope}
\node [anchor=west] (s1) at (0,0) {��\footnotesize{$_1$}};
\node [anchor=west] (s2) at ([xshift=0.5em]s1.east) {��\footnotesize{$_2$}};
\node [anchor=west] (s3) at ([xshift=0.5em]s2.east) {��\footnotesize{$_3$}};
\node [anchor=west] (s4) at ([xshift=0.5em]s3.east) {�е�\footnotesize{$_4$}};
\node [anchor=west] (s5) at ([xshift=0.5em]s4.east) {����\footnotesize{$_5$}};
\node [anchor=east] (s) at (s1.west) {$s=$};
\end{scope}

\begin{scope}[yshift=-2.6em]
\node [anchor=west] (t1) at (0.35em,0) {I\footnotesize{$_1$}};
\node [anchor=west] (t2) at ([xshift=0.3em,yshift=-0.1em]t1.east) {am\footnotesize{$_2$}};
\node [anchor=west] (t3) at ([xshift=0.3em,yshift=0.1em]t2.east) {satisfied\footnotesize{$_3$}};
\node [anchor=west] (t4) at ([xshift=0.3em]t3.east) {with\footnotesize{$_4$}};
\node [anchor=west] (t5) at ([xshift=0.3em,yshift=-0.2em]t4.east) {you\footnotesize{$_5$}};
\node [anchor=east] (t) at ([xshift=-0.3em]t1.west) {$t'=$};
\end{scope}


\draw [-,thick,ublue,dashed] (s1.south) -- (t1.north);
\draw [-,thick,ublue,dashed] (s4.south) -- ([yshift=0.3em]t2.north);
\draw [-,thick,ublue,dashed] (s2.south) ..controls +(south:1em) and +(north:1em).. (t4.north);
\draw [-,thick,ublue,dashed] (s3.south) ..controls +(south:0.5em) and +(north:1.5em).. (t5.north);
\draw [-,thick,ublue,dashed] (s5.south) -- (t3.north);
}

\end{scope}

\end{tikzpicture}

&amp; \tikz{\node[minimum height=3.2em]{\small{0.0023}\only&lt;2-&gt;{\alert{\small{$\times$0.0107}}}};} \\

\begin{tikzpicture}

\begin{scope}

{\footnotesize
\begin{scope}
\node [anchor=west] (s1) at (0,0) {��\footnotesize{$_1$}};
\node [anchor=west] (s2) at ([xshift=0.5em]s1.east) {��\footnotesize{$_2$}};
\node [anchor=west] (s3) at ([xshift=0.5em]s2.east) {��\footnotesize{$_3$}};
\node [anchor=west] (s4) at ([xshift=0.5em]s3.east) {�е�\footnotesize{$_4$}};
\node [anchor=west] (s5) at ([xshift=0.5em]s4.east) {����\footnotesize{$_5$}};
\node [anchor=east] (s) at (s1.west) {$s=$};
\end{scope}

\begin{scope}[yshift=-2.6em]
\node [anchor=center] (t1) at ([yshift=-1.6em]s1.south) {I\footnotesize{$_1$}};
\node [anchor=center] (t2) at ([yshift=-1.6em]s2.south) {with\footnotesize{$_2$}};
\node [anchor=center] (t3) at ([yshift=-1.7em]s3.south) {you\footnotesize{$_3$}};
\node [anchor=center] (t4) at ([yshift=-1.7em]s4.south) {am\footnotesize{$_4$}};
\node [anchor=center] (t5) at ([yshift=-1.6em]s5.south) {satisfied\footnotesize{$_5$}};
\node [anchor=center] (t) at ([xshift=-1.3em]t1.west) {$t''=$};
\end{scope}


\draw [-,thick,ublue,dashed] (s1.south) -- (t1.north);
\draw [-,thick,ublue,dashed] (s2.south) -- (t2.north);
\draw [-,thick,ublue,dashed] (s3.south) -- (t3.north);
\draw [-,thick,ublue,dashed] (s4.south) -- (t4.north);
\draw [-,thick,ublue,dashed] (s5.south) -- (t5.north);
}

\end{scope}

\end{tikzpicture}

&amp; \tikz{\node[minimum height=3em]{\small{0.0023}\only&lt;2-&gt;{\alert{\small{$\times$0.0009}}}};}

\end{tabular}

\item&lt;2-&gt; \textbf{�������}����������ģ��$\textrm{P}_{lm}(t)$���������ĵ�������

$\textrm{P}_{\textrm{2-gram}}(w_1...w_m)=\textrm{P}(w_1) \times \textrm{P}(w_2 | w_1) \times \textrm{P}(w_3 | w_2) ... \times \textrm{P}(w_m | w_{m-1})$

\item&lt;2-&gt; \textbf{����}

\vspace{-1em}

\begin{displaymath}
g(s,t)=\prod\nolimits_{(j,i) \in \hat{A}} \textrm{P}(s_j,t_i) \times \textrm{P}_{lm}(t)
\end{displaymath}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 3 decoding - problem definition
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ������}

\begin{center}
\begin{tikzpicture}

{\scriptsize
\node [anchor=north west,inner sep=1pt] (entry1) at (0,0) {\tiny{\textbf{1:} �� �� ���� $\leftrightarrow$ This is data}};
\node [anchor=north west,inner sep=1pt] (entry2) at ([yshift=0.1em]entry1.south west) {\tiny{\textbf{2:} С�� ��$\leftrightarrow$ Look out !}};
\node [anchor=north west,inner sep=1pt] (entry3) at ([yshift=0.1em]entry2.south west) {\tiny{\textbf{3:} �� �� ˭ $\leftrightarrow$ Who are you}};
\node [anchor=north west,inner sep=2pt] (entry4) at ([yshift=0.1em]entry3.south west) {...};
\node [anchor=south west] (corpuslabel) at (entry1.north west) {{\color{ublue} \textbf{˫��ƽ������}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (entry1) (entry2) (entry3) (entry4) (corpuslabel)] (corpus) {};
\end{pgfonlayer}
}

\node [anchor=west,ugreen] (P) at ([xshift=4em,yshift=-0.7em]corpus.east){P($t|s$)};
\node [anchor=south] (modellabel) at (P.north) {{\color{ublue} {\scriptsize \textbf{����ģ��}}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (P) (modellabel)] (model) {};
\end{pgfonlayer}

\draw [-&gt;,very thick,ublue] ([xshift=0.2em]corpus.east) -- ([xshift=3.0em]corpus.east)  node [inner sep=0pt,pos=0.5,above,yshift=0.3em] (trainingarrow) {\alert{\scriptsize{ģ��ѧϰ}}};

\draw [-&gt;,very thick,ublue] ([xshift=0.4em]model.east) -- ([xshift=3.4em]model.east)  node [inner sep=0pt,pos=0.5,above,yshift=0.3em] (decodingarrow) {\alert{\scriptsize{���\&amp;����}}};

{\scriptsize
\node [anchor=north west,inner sep=2pt] (sentlabel) at ([xshift=5.5em,yshift=-0.3em]model.north east) {{\color{ublue} \textbf{������������}}};
\node [anchor=north west] (sent) at ([yshift=-0.5em]sentlabel.south west) {\textbf{���������}};
\node [anchor=north west] (sentpart2) at ([yshift=0.3em]sent.south west) {\textbf{���з���}};
}

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (sentlabel) (sent) (sentpart2)] (segsystem) {};
\end{pgfonlayer}

\node[rectangle,fill=white,fill opacity=0.85,inner sep=0pt] [fit = (corpus) (trainingarrow)] (corpus2) {};

\end{tikzpicture}
\end{center}

\begin{itemize}
\item \textbf{����3������ - }�������$s$���ҵ����������������$\hat{t}$
\begin{displaymath}
\hat{t} = \argmax_{t} \textrm{P}(t|s)
\end{displaymath}

����$\argmax_{a} f(a)$��ʾ�ҵ�ʹ$f(a)$�ﵽ����$a$���

\item&lt;2-&gt; �������ǿ��Զ������$(s,t)$����$\textrm{P}(t|s) = \frac{g(s,t)}{\sum_{t'}g(s,t')}$
      \begin{itemize}
      \item ����$s$��$\sum_{t'}g(s,t')$�Ǹ�����(��Ϊ$\sum_{t'}g(s,t')$�ı���ֻ��$s$)
      \item \textbf{���������ǵõ����벽�����ʽ������Ϊ}
      \end{itemize}


\vspace{-1em}

\begin{eqnarray}
\hat{t} &amp; = &amp; \argmax_{t} \frac{g(s,t)}{\sum_{t'}g(s,t')} \nonumber \\
        &amp; =      &amp; \argmax_{t} g(s,t) \nonumber
\end{eqnarray}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 3 decoding - implementation
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ������(2)}
\vspace{0.5em}
\begin{itemize}
\item \textbf{����ĺ�������}�������п��ܵķ��������ҵ�ʹ$g(s,t)$�ﵽ��������\\
\vspace{-1em}
\begin{minipage}[t]{0.58\linewidth}
\begin{itemize}
\item ��$s$��$m$���ʣ�ÿ������$n$�������ѡ - ����$n^m$�����
\vspace{-0.5em}
\item&lt;2-&gt; �ʵķ����ѡ�����������
\vspace{-0.5em}
\item&lt;3-&gt; $s$��Ӧ���ܵ�����������$n^m \cdot m!$
\end{itemize}
\end{minipage}
\hfill
\begin{minipage}[t]{0.40\linewidth}
\vspace{-0.5em}
\begin{center}
\begin{tikzpicture}

\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,red] (s1) at ([yshift=-1.5em]sent.south west){\tiny{1}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,ugreen] (s2) at ([xshift=0.4em]s1.east) {\tiny{2}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,orange] (s3) at ([xshift=0.4em]s2.east) {\tiny{3}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.6pt,ublue] (s4) at ([xshift=0.4em]s3.east) {.};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=1.5pt,purple] (s5) at ([xshift=0.4em]s4.east) {\tiny{$m$}};

\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,red,fill=red] (t1) at ([yshift=-1.7em]s1.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ugreen,fill=ugreen] (t2) at ([yshift=-1.7em]s2.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,orange,fill=orange] (t3) at ([yshift=-1.7em]s3.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ublue,fill=ublue] (t4) at ([yshift=-1.7em]s4.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,purple,fill=purple] (t5) at ([yshift=-1.7em]s5.center) {{\color{white} \tiny{$n$}}};

\visible&lt;2-&gt;{
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,purple,fill=purple] (tt1) at ([yshift=-1.2em]t1.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ublue,fill=ublue] (tt2) at ([yshift=-1.2em]t2.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,orange,fill=orange] (tt3) at ([yshift=-1.2em]t3.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ugreen,fill=ugreen] (tt4) at ([yshift=-1.2em]t4.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,red,fill=red] (tt5) at ([yshift=-1.2em]t5.center) {{\color{white} \tiny{$n$}}};
}

\draw [-&gt;,thick] ([yshift=-0.1em]s1.south) -- ([yshift=0.1em]t1.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s2.south) -- ([yshift=0.1em]t2.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s3.south) -- ([yshift=0.1em]t3.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s4.south) -- ([yshift=0.1em]t4.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s5.south) -- ([yshift=0.1em]t5.north);

\visible&lt;3-&gt;{
\node [anchor=center,rotate=90] (dots) at ([yshift=-1.1em]tt3.center) {\footnotesize{...}};

\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,purple,fill=purple] (ttt1) at ([yshift=-2.2em]tt1.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,red,fill=red] (ttt2) at ([yshift=-2.2em]tt2.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ugreen,fill=ugreen] (ttt3) at ([yshift=-2.2em]tt3.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,orange,fill=orange] (ttt4) at ([yshift=-2.2em]tt4.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ublue,fill=ublue] (ttt5) at ([yshift=-2.2em]tt5.center) {{\color{white} \tiny{$n$}}};

\draw[decorate,thick,decoration={brace,amplitude=5pt,mirror}] ([yshift=-0em,xshift=0.3em]ttt5.south east) -- ([yshift=-0em,xshift=0.3em]t5.north east) node [pos=0.5,right,xshift=0.3em,yshift=0.2em] (label3) {\scriptsize{\textbf{$m!$}}};
}

\end{tikzpicture}
\end{center}
\end{minipage}

\item&lt;4-&gt; Ǳ�ڷ����������޴��޷�������������Ҫ\alert{��Ч}�������㷨�ҵ�����Ľ�

\begin{center}

\begin{tabular}{l | l | l}
$m$ &amp; $n$ &amp; $n^m \cdot m!$ \\ \hline
1 &amp; 1 &amp; 1 \\
1 &amp; 10 &amp; 10 \\
2 &amp; 10 &amp; 200 \\
10 &amp; 10 &amp; 36288000000000000 \\
20 &amp; 10 &amp; 2.43290200817664 $\times 10^{38}$ \\
20 &amp; 30 &amp; 8.48300477127188 $\times 10^{47}$ \\
\end{tabular}

\end{center}

\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 3 decoding - algorithm
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ������(3)}
\label{simpledecodingalgorithm}
��������ʹ��һ��\alert{̰��}�Ľ����㷨��ÿ��ѡ��һ��Դ���Ե��ʣ��������ķ����ѡ��չ���ģ�����"���"�Ľ������������ֱ������Դ���Ե��ʶ�������

\vspace{-1.5em}

\begin{flushleft}
\begin{tikzpicture}

{\scriptsize

\node [anchor=north west,inner sep=2pt,align=left] (line1) at (0,0) {\textrm{\textbf{Function} \textsc{WordDecoding}($s$)}};
\node [anchor=north west,inner sep=2pt,align=left] (line2) at ([yshift=-1pt]line1.south west) {\textrm{1: $\pi = $\textsc{GetTransOptions}($s$)}};
\node [anchor=north west,inner sep=2pt,align=left] (line3) at ([yshift=-1pt]line2.south west) {\textrm{2: $best = \phi$}};
\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] (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} $used[best.j] = \textrm{\textsc{\textbf{true}}}$}};
\node [anchor=north west,inner sep=2pt,align=left] (line11) at ([yshift=-1pt]line10.south west) {\textrm{10: \textbf{return} $best.translatoin$}};

\node [anchor=south west,inner sep=2pt,align=left] (head1) at ([yshift=1pt]line1.north west) {���: �ҵ��������};
\node [anchor=south west,inner sep=2pt,align=left] (head2) at ([yshift=1pt]head1.north west) {����: Դ�����$s=s_1...s_m$};

}

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (head2) (line8) (line11)] (algorithm) {};

%% highlights
%\begin{pgfonlayer}{background}
\visible&lt;2-&gt;{
\node[anchor=west,fill=blue!20,minimum height=0.16in,minimum width=1.8in] (line2highlight) at (line2.west) {};
}
\visible&lt;3-&gt;{
\node[anchor=west,fill=blue!20,minimum height=0.16in,minimum width=1.8in] (line3highlight) at (line3.west) {};
\node[anchor=west,fill=blue!20,minimum height=0.16in,minimum width=1.8in] (line5highlight) at (line5.west) {};
}
\visible&lt;4-&gt;{
\node[anchor=west,fill=blue!20,minimum height=0.16in,minimum width=1.8in] (line8highlight) at (line8.west) {};
}
\visible&lt;5-&gt;{
\node[anchor=west,fill=blue!20,minimum height=0.16in,minimum width=1.8in] (line9highlight) at (line9.west) {};
}
\visible&lt;6-&gt;{
\node[anchor=west,fill=blue!20,minimum height=0.16in,minimum width=1.8in] (line10highlight) at (line10.west) {};
}
%\end{pgfonlayer}

\end{pgfonlayer}

{\footnotesize

%% remark 1
\begin{scope}
\visible&lt;2-&gt;{
\node [anchor=north west,align=left] (remark1) at ([xshift=0.2in]algorithm.north east) {��ȡÿ������\\�ķ����ѡ};

\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,red] (s1) at ([yshift=-1.0em,xshift=0.5em]remark1.north east){\tiny{1}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,ugreen] (s2) at ([xshift=0.4em]s1.east) {\tiny{2}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,orange] (s3) at ([xshift=0.4em]s2.east) {\tiny{3}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=3.0pt,ublue] (s4) at ([xshift=0.4em]s3.east) {.};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=1.5pt,purple] (s5) at ([xshift=0.4em]s4.east) {\tiny{$m$}};

\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,red,fill=red] (t1) at ([yshift=-1.7em]s1.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ugreen,fill=ugreen] (t2) at ([yshift=-1.7em]s2.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,orange,fill=orange] (t3) at ([yshift=-1.7em]s3.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,ublue,fill=ublue] (t4) at ([yshift=-1.7em]s4.center) {{\color{white} \tiny{$n$}}};
\node [anchor=center,draw,thick,circle,minimum size=0.3em,inner sep=2pt,purple,fill=purple] (t5) at ([yshift=-1.7em]s5.center) {{\color{white} \tiny{$n$}}};

\draw [-&gt;,thick] ([yshift=-0.1em]s1.south) -- ([yshift=0.1em]t1.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s2.south) -- ([yshift=0.1em]t2.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s3.south) -- ([yshift=0.1em]t3.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s4.south) -- ([yshift=0.1em]t4.north);
\draw [-&gt;,thick] ([yshift=-0.1em]s5.south) -- ([yshift=0.1em]t5.north);

\begin{pgfonlayer}{background}
\visible&lt;2-&gt;{
\node[rectangle,draw,inner sep=0.2em,fill=blue!10] [fit = (remark1) (t5)] (remark1label) {};
}
\end{pgfonlayer}
}
\end{scope}
%% end of remark 1

%% remark 2
\begin{scope}
\visible&lt;3-&gt;{
\node [anchor=north west,draw,inner sep=2pt,fill=blue!10] (remark2) at ([xshift=-0.2em,yshift=-1em]remark1.south west) {$best$���ڱ��浱ǰ��õķ�����};
}
\end{scope}
%% end of remark 2

%% remark 3
\begin{scope}
\visible&lt;3-&gt;{
\node [anchor=north west,draw,inner sep=2pt,fill=blue!10] (remark3) at ([yshift=-0.5em]remark2.south west) {$h$���ڱ���ÿ�����ɵ��������ĺ�ѡ};
}
\end{scope}
%% end of remark 3

%% remark 4
\begin{scope}
\visible&lt;4-&gt;{
\node [anchor=north west,inner sep=2pt,align=left] (remark4) at ([xshift=0.25em,yshift=-0.6em]remark3.south west) {\textsc{Join}($a,b$) ����\\$a$ ��$b$ ���������};

{\scriptsize
\node [anchor=north west,inner sep=1pt,align=center,draw] (a1) at ([yshift=-0.5em]remark4.north east) {a1\\a2};
\node [anchor=west] (join) at (a1.east) {$\times$};
\node [anchor=north west,inner sep=1pt,align=center,draw] (b1) at ([xshift=1.5em]a1.north east) {b1\\b2};
\node [anchor=west] (join) at (b1.east) {$=$};
\node [anchor=north west,inner sep=1pt,align=center,draw] (result) at ([xshift=1.5em]b1.north east) {a1b1 a1b2\\a2b1 a2b2};
}

\begin{pgfonlayer}{background}
\visible&lt;4-&gt;{
\node[rectangle,draw,inner sep=2pt,fill=blue!10] [fit = (remark4) (result)] (remark4label) {};
}
\end{pgfonlayer}
}
\end{scope}
%% end of remark 4

%% remark 5
\begin{scope}
\visible&lt;5-&gt;{
\node [anchor=north west,align=left] (remark5) at ([xshift=0.0em,yshift=-1em]remark4.south west) {\textsc{PruneForTop1}\\�����÷���ߵĽ��};
\node [anchor=west,draw,inner sep=1pt] (s1) at ([yshift=-0.5em,xshift=1.2em]remark5.north east){\tiny{0.234}};
\node [anchor=north west,draw,inner sep=1pt] (s2) at ([yshift=-0.2em]s1.south west){\tiny{0.197}};
\node [anchor=north west,draw,inner sep=1pt] (s3) at ([yshift=-0.2em]s2.south west){\tiny{0.083}};

\draw [-] ([yshift=-0.1em,xshift=-0.2em]s1.south west) -- ([yshift=-0.1em,xshift=3em]s1.south east);
\node [anchor=west] (top1) at ([xshift=0.1em]s1.east) {\tiny{\alert{$\gets$ top1}}};

\begin{pgfonlayer}{background}
\visible&lt;5-&gt;{
\node[rectangle,draw,inner sep=0.2em,fill=blue!10] [fit = (remark5) (top1)] (remark5label) {};
}
\end{pgfonlayer}
}
\end{scope}
\%% end of remark 5

%% remark 6
\begin{scope}
\visible&lt;6-&gt;{
\node [anchor=north west,align=left] (remark6) at ([xshift=0.0em,yshift=-1em]remark5.south west) {��¼�Ѿ������\\��Դ�ﵥ��};

\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,red] (s1) at ([yshift=-1.0em,xshift=0.5em]remark6.north east){\tiny{1}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,ugreen] (s2) at ([xshift=0.4em]s1.east) {\tiny{2}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=2.1pt,orange] (s3) at ([xshift=0.4em]s2.east) {\tiny{3}};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=3.0pt,ublue] (s4) at ([xshift=0.4em]s3.east) {.};
\node [anchor=west,draw,thick,circle,minimum size=0.3em,inner sep=1.5pt,purple] (s5) at ([xshift=0.4em]s4.east) {\tiny{$m$}};

\draw [-,thick,red] (s1.north east) -- (s1.south west);
\draw [-,thick,orange] (s3.north east) -- (s3.south west);

\begin{pgfonlayer}{background}
\visible&lt;6-&gt;{
\node[rectangle,draw,inner sep=0.2em,fill=blue!10] [fit = (remark6) (s5)] (remark6label) {};
}
\end{pgfonlayer}
}
\end{scope}
\%% end of remark 6

\visible&lt;2-&gt;{
\draw [-&gt;,thick] (line2highlight.east) ..controls +(east:1em) and +(west:1em).. (remark1label.west);
}
\visible&lt;3-&gt;{
\draw [-&gt;,thick] (line3highlight.east) ..controls +(east:1em) and +(west:1em).. ([yshift=0.3em]remark2.south west);
\draw [-&gt;,thick] (line5highlight.east) ..controls +(east:1em) and +(west:1em).. ([yshift=0.3em]remark3.south west);
}
\visible&lt;4-&gt;{
\draw [-&gt;,thick] (line8highlight.east) ..controls +(east:1em) and +(west:1em).. ([yshift=0.5em]remark4label.west);
}
\visible&lt;5-&gt;{
\draw [-&gt;,thick] (line9highlight.east) ..controls +(east:1em) and +(west:1em).. ([yshift=0.5em]remark5label.west);
}
\visible&lt;6-&gt;{
\draw [-&gt;,thick] (line10highlight.east) ..controls +(east:1em) and +(west:1em).. ([yshift=0.5em]remark6label.south west);
}
}

\end{tikzpicture}
\end{flushleft}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���׻�������ϵͳ - step 3 decoding - example
\begin{frame}{ʵ��һ���򵥵Ļ�������ϵͳ������(4)}

����һ��������̵�����ʵ��

\vspace{0.5em}

\begin{tikzpicture}

\begin{scope}[xshift=-2.0in,yshift=-0.1in]

{\tiny

\node [anchor=north west,inner sep=2pt,align=left] (line1) at (0,0) {\textrm{\textbf{Function} \textsc{WordDecoding}($s$)}};
\node [anchor=north west,inner sep=2pt,align=left] (line2) at ([yshift=-3pt]line1.south west) {\textrm{1: $\pi = $\textsc{GetTransOptions}($s$)}};
\node [anchor=north west,inner sep=2pt,align=left] (line3) at ([yshift=-3pt]line2.south west) {\textrm{2: $best = \phi$}};
\node [anchor=north west,inner sep=2pt,align=left] (line4) at ([yshift=-3pt]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=-3pt]line4.south west) {\textrm{4: \hspace{1em} $h = \phi$}};
\node [anchor=north west,inner sep=2pt,align=left] (line6) at ([yshift=-3pt]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=-3pt]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=-3pt]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=-3pt]line8.south west) {\textrm{8: \hspace{1em} $best = \textrm{\textsc{PruneForTop1}}(h)$}};
\node [anchor=north west,inner sep=2pt,align=left] (line10) at ([yshift=-3pt]line9.south west) {\textrm{9: \hspace{1em} $used[best.j] = \textrm{\textsc{\textbf{true}}}$}};
\node [anchor=north west,inner sep=2pt,align=left] (line11) at ([yshift=-3pt]line10.south west) {\textrm{10: \textbf{return} $best.translatoin$}};

{\scriptsize
\node [anchor=north west,inner sep=2pt,align=left] (timeo) at ([yshift=-1em]line11.south west) {\textbf{ʱ�临�Ӷ�: } $O(m^2 \cdot n)$};
\node [anchor=north west,inner sep=2pt,align=left] (spaceo) at ([yshift=0em]timeo.south west) {\textbf{�ռ临�Ӷ�: } $O(m \cdot n)$};
}

\node [anchor=south west,inner sep=2pt,align=left] (head1) at ([yshift=3pt]line1.north west) {���: �ҵ��������};
\node [anchor=south west,inner sep=2pt,align=left] (head2) at ([yshift=3pt]head1.north west) {����: Դ�����$s=s_1...s_m$};

}

\begin{pgfonlayer}{background}
\node [rectangle,draw=ublue,thick,inner sep=0.2em,fill=white,drop shadow,minimum height=1.6cm] [fit = (head2) (line8) (line11)] (algorithm) {};

\visible&lt;2-4&gt;{
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight2) at (line2.west) {};
}
\visible&lt;5&gt;{
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight3) at (line3.west) {};
}
\visible&lt;6,12,18,25,31&gt;{
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight5) at (line5.west) {};
}
\visible&lt;7-8,13-14,19-21,26-27,32&gt;{
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight7) at (line7.west) {};
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight8) at (line8.west) {};
}
\visible&lt;9-10,15-16,22-23,28-29,33-34&gt;{
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight9) at (line9.west) {};
}
\visible&lt;11,17,24,30,35&gt;{
\node [anchor=west,fill=blue!20,minimum height=1.0em,minimum width=9.7em] (highlight10) at (line10.west) {};
}
\end{pgfonlayer}

\end{scope}

\begin{scope}[scale=0.6]

{\scriptsize

%% input words
\node [anchor=west] (s1) at (0,0) {\textbf{��}};
\node [anchor=west] (s2) at ([xshift=3em]s1.east) {\textbf{��}};
\node [anchor=west] (s3) at ([xshift=3em]s2.east) {\textbf{��}};
\node [anchor=west] (s4) at ([xshift=2.5em]s3.east) {\textbf{��ʾ}};
\node [anchor=west] (s5) at ([xshift=2.5em]s4.east) {\textbf{����}};

\node [anchor=south west,inner sep=1pt] (sentlabel) at ([yshift=0.8em]s1.north west) {\scriptsize{\textbf{����: ���������(�Ѿ��ִ�)}}};

\visible&lt;2-&gt;{
\draw [-&gt;,very thick,ublue] ([yshift=0.2em]s1.south) -- ([yshift=-0.8em]s1.south) node [pos=0.5,right] (pi1) {\tiny{$\pi$(1)}};
\draw [-&gt;,very thick,ublue] ([yshift=0.2em]s2.south) -- ([yshift=-0.8em]s2.south) node [pos=0.5,right] (pi2) {\tiny{$\pi$(2)}};
\draw [-&gt;,very thick,ublue] ([yshift=0.2em]s3.south) -- ([yshift=-0.8em]s3.south) node [pos=0.5,right] (pi3) {\tiny{$\pi$(3)}};
\draw [-&gt;,very thick,ublue] ([yshift=0.2em]s4.south) -- ([yshift=-0.8em]s4.south) node [pos=0.5,right] (pi4) {\tiny{$\pi$(4)}};
\draw [-&gt;,very thick,ublue] ([yshift=0.2em]s5.south) -- ([yshift=-0.8em]s5.south) node [pos=0.5,right] (pi5) {\tiny{$\pi$(5)}};
}

\visible&lt;11-&gt;{
\node [anchor=south] (used1) at ([yshift=-1em]s1.north) {\tiny{\alert{\textbf{used}}}};
}
\visible&lt;30-&gt;{
\node [anchor=south] (used2) at ([yshift=-1em]s2.north) {\tiny{\alert{\textbf{used}}}};
}
\visible&lt;35-&gt;{
\node [anchor=south] (used3) at ([yshift=-1em]s3.north) {\tiny{\alert{\textbf{used}}}};
}
\visible&lt;24-&gt;{
\node [anchor=south] (used4) at ([yshift=-1em]s4.north) {\tiny{\alert{\textbf{used}}}};
}
\visible&lt;17-&gt;{
\node [anchor=south] (used5) at ([yshift=-1em]s5.north) {\tiny{\alert{\textbf{used}}}};
}

}

{\tiny

%% translation options
\visible&lt;2-&gt;{
\node [anchor=north,inner sep=2pt,fill=red!20,minimum height=1.5em,minimum width=2.5em] (t11) at ([yshift=-1.3em]s1.south) {I'm};
\node [anchor=north,inner sep=2pt,fill=red!20,minimum height=1.5em,minimum width=2.5em] (t12) at ([yshift=-0.2em]t11.south) {I};
\node [anchor=north,inner sep=2pt,fill=red!20,minimum height=1.5em,minimum width=2.5em] (t13) at ([yshift=-0.2em]t12.south) {me};
}
\visible&lt;2-3&gt;{
\node [anchor=north,inner sep=2pt,fill=red!20,minimum height=1.5em,minimum width=2.5em] (t14) at ([yshift=-0.2em]t13.south) {am};
\node [anchor=north,inner sep=2pt,fill=red!20,minimum height=1.5em,minimum width=2.5em] (t15) at ([yshift=-0.2em]t14.south) {...};
}

\visible&lt;2-&gt;{
\node [anchor=north,inner sep=2pt,fill=green!20,minimum height=1.5em,minimum width=2.5em] (t21) at ([yshift=-1.3em]s2.south) {to};
\node [anchor=north,inner sep=2pt,fill=green!20,minimum height=1.5em,minimum width=2.5em] (t22) at ([yshift=-0.2em]t21.south) {with};
\node [anchor=north,inner sep=2pt,fill=green!20,minimum height=1.5em,minimum width=2.5em] (t23) at ([yshift=-0.2em]t22.south) {for};
}
\visible&lt;2-3&gt;{
\node [anchor=north,inner sep=2pt,fill=green!20,minimum height=1.5em,minimum width=2.5em] (t24) at ([yshift=-0.2em]t23.south) {to};
\node [anchor=north,inner sep=2pt,fill=green!20,minimum height=1.5em,minimum width=2.5em] (t25) at ([yshift=-0.2em]t24.south) {...};
}

\visible&lt;2-&gt;{
\node [anchor=north,inner sep=2pt,fill=blue!20,minimum height=1.5em,minimum width=2.5em] (t31) at ([yshift=-1.3em]s3.south) {you};
\node [anchor=north,inner sep=2pt,fill=blue!20,minimum height=1.5em,minimum width=2.5em] (t32) at ([yshift=-0.2em]t31.south) {your};
}

\visible&lt;2-&gt;{
\node [anchor=north,inner sep=2pt,fill=orange!20,minimum height=1.5em,minimum width=3.5em] (t41) at ([yshift=-1.3em]s4.south) {$\phi$};
\node [anchor=north,inner sep=2pt,fill=orange!20,minimum height=1.5em,minimum width=3.5em] (t42) at ([yshift=-0.2em]t41.south) {show};
\node [anchor=north,inner sep=2pt,fill=orange!20,minimum height=1.5em,minimum width=3.5em] (t43) at ([yshift=-0.2em]t42.south) {shows};
}
\visible&lt;2-3&gt;{
\node [anchor=north,inner sep=2pt,fill=orange!20,minimum height=1.5em,minimum width=3.5em] (t44) at ([yshift=-0.2em]t43.south) {means};
\node [anchor=north,inner sep=2pt,fill=orange!20,minimum height=1.5em,minimum width=3.5em] (t45) at ([yshift=-0.2em]t44.south) {...};
}

\visible&lt;2-&gt;{
\node [anchor=north,inner sep=2pt,fill=purple!20,minimum height=1.5em,minimum width=4.5em] (t51) at ([yshift=-1.3em]s5.south) {satisfy};
\node [anchor=north,inner sep=2pt,fill=purple!20,minimum height=1.5em,minimum width=4.5em] (t52) at ([yshift=-0.2em]t51.south) {satisfied};
\node [anchor=north,inner sep=2pt,fill=purple!20,minimum height=1.5em,minimum width=4.5em] (t53) at ([yshift=-0.2em]t52.south) {satisfies};
}
\visible&lt;2-3&gt;{
\node [anchor=north,inner sep=2pt,fill=purple!20,minimum height=1.5em,minimum width=4.5em] (t54) at ([yshift=-0.2em]t53.south) {it};
\node [anchor=north,inner sep=2pt,fill=purple!20,minimum height=1.5em,minimum width=4.5em] (t55) at ([yshift=-0.2em]t54.south) {...};
}

\visible&lt;2-&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt11) at (t11.east) {{\color{white} .4}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt12) at (t12.east) {{\color{white} .3}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt13) at (t13.east) {{\color{white} .1}};
}
\visible&lt;2-3&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt14) at (t14.east) {{\color{white} .1}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,minimum height=0.95em,fill=black] (pt15) at (t15.east) {{\color{white} ...}};
}

\visible&lt;2-&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt21) at (t21.east) {{\color{white} .3}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt22) at (t22.east) {{\color{white} .3}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt23) at (t23.east) {{\color{white} .2}};
}
\visible&lt;2-3&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt24) at (t24.east) {{\color{white} .1}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,minimum height=0.95em,fill=black] (pt25) at (t25.east) {{\color{white} ...}};
}

\visible&lt;2-&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt31) at (t31.east) {{\color{white} .7}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt32) at (t32.east) {{\color{white} .3}};
}

\visible&lt;2-&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt41) at (t41.east) {{\color{white} .4}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt42) at (t42.east) {{\color{white} .2}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt43) at (t43.east) {{\color{white} .1}};
}
\visible&lt;2-3&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt44) at (t44.east) {{\color{white} .1}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,minimum height=0.95em,fill=black] (pt45) at (t45.east) {{\color{white} ...}};
}

\visible&lt;2-&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt51) at (t51.east) {{\color{white} .3}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt52) at (t52.east) {{\color{white} .2}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt53) at (t53.east) {{\color{white} .2}};
}
\visible&lt;2-3&gt;{
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,fill=black] (pt54) at (t54.east) {{\color{white} .1}};
\node [anchor=north,rotate=90,inner sep=1pt,minimum width=1.5em,minimum height=0.95em,fill=black] (pt55) at (t55.east) {{\color{white} ...}};
}

}

\visible&lt;3&gt;{
\draw [-] ([yshift=-0.5pt,xshift=-0.5em]t13.south west) -- ([yshift=-0.5pt,xshift=1.5em]t53.south east);
}
\visible&lt;4&gt;{
\node [anchor=center,rotate=90] (tablepruninglabel) at ([xshift=-1.5em]t12.west) {\tiny{\alert{ֻ����ǰ$n$�����}}};
}


{\scriptsize

%% 1-best hypothesis
\visible&lt;5-&gt;{
\node [anchor=west] (translation) at ([yshift=-4em]t13.south west) {$best.translation = $};
}
\visible&lt;5-9&gt;{
\node [anchor=west,inner sep=2pt,minimum height=1.5em,minimum width=2em] (transword0) at (translation.east) {$\phi$};
}
\visible&lt;10-&gt;{
\node [anchor=west,fill=red!20,inner sep=2pt,minimum height=1.5em,minimum width=2em] (transword1) at (translation.east) {I'm};
}
\visible&lt;16-&gt;{
\node [anchor=west,fill=purple!20,inner sep=2pt,minimum height=1.5em] (transword2) at ([xshift=2pt]transword1.east) {satisfied};
}
\visible&lt;29-&gt;{
\node [anchor=west,fill=green!20,inner sep=2pt,minimum height=1.5em] (transword3) at ([xshift=2pt]transword2.east) {with};
}
\visible&lt;34-&gt;{
\node [anchor=west,fill=blue!20,inner sep=2pt,minimum height=1.5em] (transword4) at ([xshift=2pt]transword3.east) {you};
}

\visible&lt;7-8,13-14&gt;{
\node [anchor=west,draw,inner sep=0pt,minimum height=1.8em,minimum width=2.3em] (hypotrans1) at ([xshift=-0.2em]transword1.west) {};
}
\visible&lt;19-21,26-27&gt;{
\node [anchor=west,draw,inner sep=0pt,minimum height=1.8em,minimum width=6.20em] (hypotrans2) at ([xshift=-0.2em]transword1.west) {};
}
\visible&lt;32&gt;{
\node [anchor=west,draw,inner sep=0pt,minimum height=1.8em,minimum width=8.50em] (hypotrans3) at ([xshift=-0.2em]transword1.west) {};
}

\visible&lt;5-9&gt;{
\node [anchor=north west] (jlabel) at (translation.south west) {$best.j = -1$};
}
\visible&lt;10-15&gt;{
\node [anchor=north west] (jlabel) at (translation.south west) {$best.j = 1$};
}
\visible&lt;16-22&gt;{
\node [anchor=north west] (jlabel) at (translation.south west) {$best.j = 5$};
}
\visible&lt;23-28&gt;{
\node [anchor=north west] (jlabel) at (translation.south west) {$best.j = 4$};
}
\visible&lt;29-33&gt;{
\node [anchor=north west] (jlabel) at (translation.south west) {$best.j = 2$};
}
\visible&lt;34-&gt;{
\node [anchor=north west] (jlabel) at (translation.south west) {$best.j = 3$};
}

}

%% translation option lists
\visible&lt;7&gt;{
\node[anchor=west,draw,minimum height=2.8em,minimum width=2.1em] (list1) at ([xshift=-2pt]t12.west) {};
}
\visible&lt;8,13,19,26&gt;{
\node[anchor=west,draw,minimum height=2.8em,minimum width=2.1em] (list2) at ([xshift=-2pt]t22.west) {};
}
\visible&lt;14&gt;{
\node[anchor=west,draw,minimum height=2.8em,minimum width=3.2em] (list5) at ([xshift=-2pt]t52.west) {};
}
\visible&lt;20,27,32&gt;{
\node[anchor=west,draw,minimum height=1.85em,minimum width=2.1em] (list3) at ([xshift=-2pt,yshift=1pt]t32.north west) {};
}
\visible&lt;21&gt;{
\node[anchor=west,draw,minimum height=2.8em,minimum width=2.6em] (list4) at ([xshift=-2pt]t42.west) {};
}

%% translaiton hypotheses (h)
{\tiny

\visible&lt;7&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 1, j = 1$}};
}
\visible&lt;8&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 1, j = 2$}};
}
\visible&lt;9-12&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 1, j = 5$}};
}
\visible&lt;13&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 2, j = 2$}};
}
\visible&lt;14-18&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 2, j = 5$}};
}
\visible&lt;19&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 3, j = 2$}};
}
\visible&lt;20&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 3, j = 3$}};
}
\visible&lt;21-25&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 3, j = 4$}};
}
\visible&lt;26&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 4, j = 2$}};
}
\visible&lt;27-31&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 4, j = 3$}};
}
\visible&lt;32-&gt;{
\node [anchor=west] (hlabel) at ([yshift=-5em]jlabel.west) {\scriptsize{$i = 5, j = 3$}};
}

\visible&lt;6-&gt;{
\node [anchor=north west] (glabel) at (hlabel.south west) {$g(s,t)$};
\node [anchor=west] (translabel) at (glabel.east) {������};
\draw [-] (glabel.north east) -- ([yshift=-1.9in]glabel.north east);
\draw [-] (glabel.south west) -- ([xshift=3.5in]glabel.south west);

\node [anchor=center,rotate=90] (hlabel2) at ([xshift=-1.5em,yshift=-8em]glabel.west) {\tiny{$h$�����ʱ������}};
}

%% the 'join' operation
\visible&lt;7-8,13-14,19-21,26-27,32&gt;{
\node [anchor=center,draw=ublue,circle,thick,fill=white,inner sep=2.5pt,circular drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (join) at ([xshift=8em,yshift=-1em]hlabel.north east) {\scriptsize{\textsc{Join}}};
}

%% round 1
\visible&lt;7-11&gt;{
\node [anchor=north west] (g1) at ([yshift=-3pt]glabel.south west) {.1744};
\node [anchor=north west] (g2) at ([yshift=-1pt]g1.south west) {.1674};
\node [anchor=north west] (g3) at ([yshift=-1pt]g2.south west) {.1041};
}
\visible&lt;8-11&gt;{
\node [anchor=north west] (g4) at ([yshift=-1pt]g3.south west) {.0561};
\node [anchor=north west] (g5) at ([yshift=-1pt]g4.south west) {.0013};
\node [anchor=north west] (g6) at ([yshift=-1pt]g5.south west) {.0013};
}
\visible&lt;9-11&gt;{
\node [anchor=north west] (g7) at ([yshift=-1pt]g6.south west) {...};
\node [anchor=north west] (g8) at ([yshift=-1pt]g7.south west) {.1452};
}

\visible&lt;7-11&gt;{
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=red!20] (hword11) at ([xshift=2.0em]g1.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=red!20] (hword21) at ([xshift=2.0em]g2.east) {I};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=red!20] (hword31) at ([xshift=2.0em]g3.east) {me};
}
\visible&lt;8-11&gt;{
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword41) at ([xshift=2.0em]g4.east) {to};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword51) at ([xshift=2.0em]g5.east) {with};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword61) at ([xshift=2.0em]g6.east) {for};
}
\visible&lt;9-11&gt;{
\node [anchor=west,inner sep=2pt,minimum height=1.3em] (hword71) at ([xshift=4.0em]g7.east) {...};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=purple!20] (hword81) at ([xshift=2.0em]g8.east) {satisfies};
}

\visible&lt;7-8&gt;{
\draw [-&gt;,thick] (hypotrans1.south) ..controls +(south:0.5) and +(north:0.5).. (join.north);
}
\visible&lt;7&gt;{
\draw [-&gt;,thick] (list1.south) ..controls +(300:3) and +(north west:3).. (join.north west);
\draw [-&gt;,thick] (join.south) ..controls +(south:1) and +(east:1).. ([xshift=7em]g2.east);
}
\visible&lt;8&gt;{
\draw [-&gt;,thick] (list2.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south) ..controls +(south:3) and +(east:1).. ([xshift=7em]g5.east);
}

\visible&lt;9-11&gt;{
\node [anchor=west] (top1) at (hword11.east) {\alert{$\gets$ top1}};
}

%% round 2
\visible&lt;13-17&gt;{
\node [anchor=north west] (g1) at ([yshift=-3pt]glabel.south west) {.0099};
\node [anchor=north west] (g2) at ([yshift=-1pt]g1.south west) {.0300};
\node [anchor=north west] (g3) at ([yshift=-1pt]g2.south west) {.0231};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword11) at ([xshift=2.0em]g1.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword21) at ([xshift=2.0em]g2.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword31) at ([xshift=2.0em]g3.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword12) at ([xshift=0.2em]hword11.east) {to};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword22) at ([xshift=0.2em]hword21.east) {with};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword32) at ([xshift=0.2em]hword31.east) {for};
}
\visible&lt;14-17&gt;{
\node [anchor=north west] (g4) at ([yshift=-1pt]g3.south west) {...};
\node [anchor=north west] (g5) at ([yshift=-1pt]g4.south west) {.0121};
\node [anchor=north west] (g6) at ([yshift=-1pt]g5.south west) {.0314};
\node [anchor=north west] (g7) at ([yshift=-1pt]g6.south west) {.0081};
\node [anchor=west,inner sep=2pt,minimum height=1.3em] (hword41) at ([xshift=4.0em]g4.east) {...};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword51) at ([xshift=2.0em]g5.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword61) at ([xshift=2.0em]g6.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword71) at ([xshift=2.0em]g7.east) {I'm};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=purple!20] (hword52) at ([xshift=0.2em]hword51.east) {satisfy};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=purple!20] (hword62) at ([xshift=0.2em]hword61.east) {satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=purple!20] (hword72) at ([xshift=0.2em]hword71.east) {satisfies};
}
\visible&lt;15-17&gt;{
\node [anchor=west] (top1) at (hword62.east) {\alert{$\gets$ top1}};
}
\visible&lt;13-14&gt;{
\draw [-&gt;,thick] (hypotrans1.south) ..controls +(south:0.5) and +(north:0.5).. (join.north);
}
\visible&lt;13&gt;{
\draw [-&gt;,thick] (list2.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south) ..controls +(south:1) and +(east:1).. ([xshift=1em]hword22.east);
}
\visible&lt;14&gt;{
\draw [-&gt;,thick] (list5.south) ..controls +(200:3) and +(20:3).. (join.north east);
\draw [-&gt;,thick] (join.south) ..controls +(south:2) and +(east:0.5).. ([xshift=1em]hword62.east);
}

%% round 3
\visible&lt;19-24&gt;{
\node [anchor=north west] (g1) at ([yshift=-3pt]glabel.south west) {.0082};
\node [anchor=north west] (g2) at ([yshift=-1pt]g1.south west) {.0103};
\node [anchor=north west] (g3) at ([yshift=-1pt]g2.south west) {.0073};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword11) at ([xshift=2.0em]g1.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword21) at ([xshift=2.0em]g2.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword31) at ([xshift=2.0em]g3.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword12) at ([xshift=0.2em]hword11.east) {to};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword22) at ([xshift=0.2em]hword21.east) {with};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword32) at ([xshift=0.2em]hword31.east) {for};
}
\visible&lt;20-24&gt;{
\node [anchor=north west] (g4) at ([yshift=-1pt]g3.south west) {.0033};
\node [anchor=north west] (g5) at ([yshift=-1pt]g4.south west) {.0039};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword41) at ([xshift=2.0em]g4.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword51) at ([xshift=2.0em]g5.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=blue!20] (hword42) at ([xshift=0.2em]hword41.east) {you};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=blue!20] (hword52) at ([xshift=0.2em]hword51.east) {your};
}
\visible&lt;21-24&gt;{
\node [anchor=north west] (g6) at ([yshift=-1pt]g5.south west) {.0106};
\node [anchor=north west] (g7) at ([yshift=-1pt]g6.south west) {.0035};
\node [anchor=north west] (g8) at ([yshift=-1pt]g7.south west) {.0028};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword61) at ([xshift=2.0em]g6.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword71) at ([xshift=2.0em]g7.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword81) at ([xshift=2.0em]g8.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=orange!20] (hword62) at ([xshift=0.2em]hword61.east) {$\phi$};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=orange!20] (hword72) at ([xshift=0.2em]hword71.east) {show};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=orange!20] (hword82) at ([xshift=0.2em]hword81.east) {shows};
}
\visible&lt;22-24&gt;{
\node [anchor=west] (top1) at (hword62.east) {\alert{$\gets$ top1}};
}
\visible&lt;19-21&gt;{
\draw [-&gt;,thick] (hypotrans2.south) ..controls +(south:0.5) and +(north:0.5).. (join.north);
}
\visible&lt;19&gt;{
\draw [-&gt;,thick] ([xshift=-2em]list2.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south east) ..controls +(330:0.5) and +(30:0.3).. ([xshift=1em]hword22.east);
}
\visible&lt;20&gt;{
\draw [-&gt;,thick] ([xshift=-2em]list3.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south east) ..controls +(330:1) and +(30:0.3).. ([xshift=1em]hword42.south east);
}
\visible&lt;21&gt;{
\draw [-&gt;,thick] (list4.south) ..controls +(south:1) and +(north:3).. (join.north west);
\draw [-&gt;,thick] (join.south east) ..controls +(330:1) and +(30:0.3).. ([xshift=1em]hword72.south east);
}

%% round 4
\visible&lt;26-30&gt;{
\node [anchor=north west] (g1) at ([yshift=-3pt]glabel.south west) {.0032};
\node [anchor=north west] (g2) at ([yshift=-1pt]g1.south west) {.0040};
\node [anchor=north west] (g3) at ([yshift=-1pt]g2.south west) {.0028};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword11) at ([xshift=2.0em]g1.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword21) at ([xshift=2.0em]g2.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword31) at ([xshift=2.0em]g3.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword12) at ([xshift=0.2em]hword11.east) {to};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword22) at ([xshift=0.2em]hword21.east) {with};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword32) at ([xshift=0.2em]hword31.east) {for};
}
\visible&lt;27-30&gt;{
\node [anchor=north west] (g4) at ([yshift=-1pt]g3.south west) {.0012};
\node [anchor=north west] (g5) at ([yshift=-1pt]g4.south west) {.0014};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword41) at ([xshift=2.0em]g4.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword51) at ([xshift=2.0em]g5.east) {I'm satisfied};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=blue!20] (hword42) at ([xshift=0.2em]hword41.east) {you};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=blue!20] (hword52) at ([xshift=0.2em]hword51.east) {your};
}
\visible&lt;28-30&gt;{
\node [anchor=west] (top1) at (hword22.east) {\alert{$\gets$ top1}};
}
\visible&lt;26-27&gt;{
\draw [-&gt;,thick] (hypotrans2.south) ..controls +(south:0.5) and +(north:0.5).. (join.north);
}
\visible&lt;26&gt;{
\draw [-&gt;,thick] ([xshift=-2em]list2.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south east) ..controls +(330:0.5) and +(30:0.3).. ([xshift=1em]hword22.east);
}
\visible&lt;27&gt;{
\draw [-&gt;,thick] ([xshift=-2em]list3.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south east) ..controls +(330:1) and +(30:0.3).. ([xshift=1em]hword42.south east);
}

%% round 5
\visible&lt;32-35&gt;{
\node [anchor=north west] (g1) at ([yshift=-3pt]glabel.south west) {.0011};
\node [anchor=north west] (g2) at ([yshift=-1pt]g1.south west) {.0010};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword11) at ([xshift=2.0em]g1.east) {I'm satisfied with};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=gray!20] (hword21) at ([xshift=2.0em]g2.east) {I'm satisfied with};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword12) at ([xshift=0.2em]hword11.east) {you};
\node [anchor=west,inner sep=2pt,minimum height=1.3em,fill=green!20] (hword22) at ([xshift=0.2em]hword21.east) {your};
}
\visible&lt;33-35&gt;{
\node [anchor=west] (top1) at (hword12.east) {\alert{$\gets$ top1}};
}
\visible&lt;32&gt;{
\draw [-&gt;,thick] (hypotrans3.south) ..controls +(south:0.5) and +(north:0.5).. (join.north);
}
\visible&lt;32&gt;{
\draw [-&gt;,thick] ([xshift=-2em]list3.south) ..controls +(south:2) and +(north:2).. (join.110);
\draw [-&gt;,thick] (join.south east) ..controls +(340:1) and +(30:0.3).. ([xshift=1em]hword22.north east);
}

}

\end{scope}

\end{tikzpicture}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ͣ�٣����ܱ�����Ҫ����
\begin{frame}{������ʼ��ʽ������}

\begin{itemize}

\item ���ڴʵ�ͳ�ƻ�������ܼ� - ����ʵ��һ��ǰ���������ķ�����ȷʵ����
    \begin{itemize}
    \item \textbf{�ǵ�}��˼��ܼ�
    \item \textbf{����}��ǰ���������Ҫ����ʵ�֣����кܶ�����û�лش�
        \begin{itemize}
        \item \textbf{��ģ��}����Ҫ�����ܵ���ѧģ���������������
        \item \textbf{ѵ����}��Ҫ��һ����ѧ���Ż�����
        \item \textbf{���룺}�Ƿ���������ϵͳ�Ľ��뷽����A*��Bottom-up decoding?
        \item \textbf{������}�շ������⡢����ģ����ȵ�
        \end{itemize}
    \end{itemize}
\item&lt;2-&gt; \textbf{����}��������һЩ������н�һ�����ܼ�����
    \begin{enumerate}
    \item ͳ�ƻ����������ģ�ͺ�IBMģ��
        \begin{itemize}
        \item ͳ�ƻ�������ģ�ͻ���(����ģ��)
        \item IBM Models 1-2�Ľ�ģ��ѵ��
        \item IBM Models 3-5�Ľ�ģ
        \end{itemize}
    \item ͳ�ƴʶ���
        \begin{itemize}
        \item ����HMMs�Ĵʶ���
        \item �����б�ʽģ�͵Ĵʶ���
        \end{itemize}
    \end{enumerate}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\section{IBMģ��}

%%%------------------------------------------------------------------------------------------------------------
\subsection{�����ŵ�ģ��}

%%%------------------------------------------------------------------------------------------------------------
%%% outline
\begin{frame}{Outline}

\begin{center}
\begin{tikzpicture}

{\Large
\node [anchor=west,draw,gray,very thick,rounded corners=4pt,text width=15em,align=left,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (p1) at (0,0) {\gray{\textbf{1. һ���򵥵ķ���ʵ��}}};

\node [anchor=north west,draw,ublue,very thick,rounded corners=4pt,text width=15em,align=left,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (p21) at ([yshift=-1em]p1.south west) {\black{\textbf{2. IBMģ��}}\\\black{\textbf{\hspace{1.5em}$\bullet$  ��ģ}}\\\gray{\textbf{\hspace{1.5em}$\bullet$  ����}}\\\gray{\textbf{\hspace{1.5em}$\bullet$  ģ��ѵ��}}};
}

\end{tikzpicture}
\end{center}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% IBM Models�ĵ�λ
\begin{frame}{ͳ�ƻ�������}

\begin{itemize}
\item ͳ�ƻ�������Դ��1941���Warren Weaver��˼�룬����ʽ��ʼ��
    {\footnotesize
    \begin{itemize}
    \item \textit{\textbf{A Program for Aligning Sentences in Bilingual Corpora}}\\
          William A. Gale; Kenneth W. Church. 1993.
    \item \textit{\textbf{The Mathematics of Statistical Machine Translation: Parameter Estimation}}\\
          Peter E. Brown; Stephen A. Della Pietra; Vincent J. Della Pietra; Robert L. Mercer. 1993.
    \end{itemize}
    }
\item �ر���IBM Waston�о�����Brown���˵Ļ��ڴʵ�ͳ�ƻ�������ģ�ͣ���Ϊ����������������\alert{�����еľ���}
\vspace{0.3em}
\begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true,width=\linewidth]{\small{Brown���˵Ĺ���(IBM Waston)} - on Wikipedia}
{\scriptsize
\begin{spacing}{1}
\textit{The first ideas of statistical machine translation were introduced by Warren Weaver in 1949, including the ideas of applying Claude Shannon's information theory. Statistical machine translation was re-introduced in 1991 by researchers at IBM's Thomas J. Watson Research Center and has contributed to the significant resurgence in interest in machine translation in recent years. Nowadays it is by far the most widely-studied machine translation method}
\vspace{-0.7em}
\end{spacing}
}
\end{beamerboxesrounded}
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ���������е�ͳ��˼��
\begin{frame}{���������ͳ�ƽ�ģ}

\begin{itemize}
\item \textbf{һ������������ʱ}�����ڸ�����Դ���Ծ���$s$�������˷���Ϊһ�����������ɸ�����ȷ������$\hat{t}$
    \begin{itemize}
    \item Ҳ����˵������ȷ�����ģ������ķ��붼�ǲ���ȷ��
    \end{itemize}

\begin{center}
\begin{tikzpicture}

\node [draw,red,fill=red!10,thick,anchor=center,circle,inner sep=3.5pt] (s) at (0,0) {\black{$s$}};
\node [draw,ublue,fill=blue!10,thick,anchor=center,circle,inner sep=2pt] (t) at ([xshift=1in]s.east) {\black{$\hat{t}$}};

\draw [-&gt;,thick,] (s.north east) .. controls +(north east:1em) and +(north west:1em).. (t.north west) node[pos=0.5,below] {\tiny{��ȷ����}};

\end{tikzpicture}
\end{center}

\item&lt;2-&gt; \textbf{ͳ�ƻ��������˼����}������$s$�����п��ܵ�Ŀ����ʴ�$t$���ǿ��ܵ����ġ�ÿһ��($s$,$t$)����һ������ֵ$\textrm{P}(t|s)$ ������$s$ ����Ϊ$t$�ĺ��뻵

\begin{center}
\begin{tikzpicture}

\node [draw,red,fill=red!10,thick,anchor=center,circle,inner sep=3.5pt] (s) at (0,0) {\black{$s$}};
\node [draw,ublue,fill=blue!10,thick,anchor=center,circle,inner sep=2pt] (t1) at ([xshift=1in]s.east) {\black{$t_1$}};
\node [draw,ublue,fill=blue!10,thick,anchor=center,circle,inner sep=2pt] (t2) at ([xshift=3em,yshift=2em]t1.north east) {\black{$t_2$}};
\node [draw,ublue,fill=blue!10,thick,anchor=center,circle,inner sep=2pt] (t3) at ([xshift=1em,yshift=4em]t1.north east) {\black{$t_3$}};
\node [draw,ublue,fill=blue!10,thick,anchor=center,circle,inner sep=2pt] (t4) at ([xshift=3em,yshift=-1.5em]t1.north east) {\black{$t_4$}};

\node [draw,dashed,ublue,fill=blue!10,thick,anchor=center,circle,minimum size=18pt] (t5) at ([xshift=3em]t3.east) {};
\node [draw,dashed,ublue,fill=blue!10,thick,anchor=center,circle,minimum size=18pt] (t6) at ([xshift=3em]t2.east) {};
\node [draw,dashed,ublue,fill=blue!10,thick,anchor=center,circle,minimum size=18pt] (t7) at ([xshift=3em]t4.east) {};

\draw [-&gt;,thick,] (s.north east) .. controls +(north east:1em) and +(north west:1em).. (t1.north west) node[pos=0.5,below] {\tiny{P ($t_1|s$)=0.1}};
\draw [-&gt;,thick,] (s.60) .. controls +(50:4em) and +(west:1em).. (t2.west) node[pos=0.5,below] {\tiny{P($t_2|s$)=0.2}};
\draw [-&gt;,thick,] (s.north) .. controls +(70:4em) and +(west:1em).. (t3.west) node[pos=0.5,above,xshift=-1em] {\tiny{P($t_3|s$)=0.3}};
\draw [-&gt;,thick,] (s.south east) .. controls +(300:3em) and +(south west:1em).. (t4.south west) node[pos=0.5,below] {\tiny{P($t_4|s$)=0.1}};

\end{tikzpicture}
\end{center}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% �����ŵ�ģ��
\begin{frame}{�����ŵ�ģ��}

\begin{itemize}
\item \textbf{�����ŵ�ģ��}��Դ���Ծ���$s$(����)����Ŀ�������$t$(��Դ)����һ�����������ŵ��õ��ġ����֪����$s$���ŵ������ʣ����ǿ���ͨ��$\textrm{P}(t|s)$�õ����ܵ���Դ�ĸ��ʡ�\\

\begin{center}
\begin{tikzpicture}

\node [draw,red,fill=red!10,thick,anchor=center,circle,inner sep=3.5pt] (s) at (0,0) {\black{$s$}};
\node [draw,ublue,fill=blue!10,thick,anchor=center,circle,inner sep=3.3pt] (t) at ([xshift=1.5in]s.east) {\black{$t$}};

\draw [&lt;-&gt;,thick,] (s.east) -- (t.west) node [pos=0.5,draw,fill=white] {�����ŵ�};
\node [anchor=east] at (s.west) {\scriptsize{����}};
\node [anchor=west] at (t.east) {\scriptsize{��Դ}};

\end{tikzpicture}
\end{center}

    ��ͨ�����������ҵ�����ܵ���Դ�Ĺ��̱���֮Ϊ\alert{����}

    \begin{displaymath}
    \hat{t} = \argmax_{t} \textrm{P}(t|s)
    \end{displaymath}

\item&lt;2-&gt; \textbf{��Ҷ˹�任}
    \begin{eqnarray}
    \textrm{P}(t|s) &amp; = &amp; \frac{\textrm{P}(s,t)}{\textrm{P}(s)} \nonumber \\
                    &amp; = &amp; \frac{\textrm{P}(s|t) \textrm{P}(t)}{\textrm{P}(s)} \nonumber
    \end{eqnarray}
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% ����ģ�� + ����ģ��
\begin{frame}{�����ŵ�ģ��(2)}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (p) at (0,0) {$\textrm{P}(t|s)$};
\node [anchor=west] (eqiv) at (p.east) {=};
\node [anchor=south west,inner sep=2pt] (transmodel) at ([yshift=-2pt]eqiv.north east) {$\textrm{P}(s|t)$};
\node [anchor=west,inner sep=2pt] (lmmodel) at ([xshift=2pt]transmodel.east) {$\textrm{P}(t)$};
\node [anchor=north west,inner sep=2pt] (sp) at ([yshift=4pt,xshift=1.5em]eqiv.south east) {$\textrm{P}(s)$};

\visible&lt;2-&gt;{
\node [anchor=south west,fill=red!20,inner sep=2pt] (transmodel) at ([yshift=-2pt]eqiv.north east) {$\textrm{P}(s|t)$};
\node [anchor=west,fill=blue!20,inner sep=2pt] (lmmodel) at ([xshift=2pt]transmodel.east) {$\textrm{P}(t)$};
\node [anchor=north west,fill=green!20,inner sep=2pt] (sp) at ([yshift=4pt,xshift=1.5em]eqiv.south east) {$\textrm{P}(s)$};
}

\draw [-] ([yshift=-4pt]transmodel.south west) -- ([yshift=-4pt]lmmodel.south east);

\visible&lt;2-&gt;{
\node [anchor=south east,fill=red!20,draw,align=left] (tmmark) at ([xshift=0.5in,yshift=0.4in]p.north) {\footnotesize{������Դ$t$���õ�����$s$ }\\\footnotesize{�ĸ��ʣ�����\textbf{����ģ��}}};

\node [anchor=west,fill=blue!20,draw,align=left] (lmmark) at ([xshift=0.5in]tmmark.east) {\footnotesize{��Դ$t$���ֵĸ���}\\\footnotesize{����\textbf{����ģ��}}};

\node [anchor=west,fill=green!20,draw,align=left] (smark) at ([xshift=0.2in,yshift=-0.3in]lmmodel.east) {\footnotesize{����$s$���ֵĸ���}\\\footnotesize{����$s$��$\textrm{P}(s)$Ϊ\textbf{����}}};

\draw [-&gt;,thick] (transmodel.north) .. controls +(north:1.5em) and + (south:1.5em) .. (tmmark.south);
\draw [-&gt;,thick] (lmmodel.north) .. controls +(north:1.5em) and + (south:1.3em) .. (lmmark.south);
\draw [-&gt;,thick] (sp.east) .. controls +(east:2em) and + (west:2em) .. ([yshift=0.3em]smark.south west);
}

\end{tikzpicture}
\end{center}

\begin{itemize}
\item&lt;3-&gt; \textbf{���}������������Ա���������Ϊ

\vspace{-0.5em}

\begin{eqnarray}
\hat{t} &amp; = &amp; \argmax_{t} \frac{\textrm{P}(s|t) \textrm{P}(t)}{\textrm{P}(s)} \nonumber \\
        &amp; = &amp; \argmax_{t} \textrm{P}(s|t) \textrm{P}(t) \nonumber
\end{eqnarray}

���������п��ܵ��������ҵ�ʹ����ģ��$\textrm{P}(s|t)$������ģ��$\textrm{P}(t)$�˻���������

\end{itemize}
\
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{������������}

%%%------------------------------------------------------------------------------------------------------------
%%% ͳ�ƻ��������������������
\begin{frame}{��������}

\begin{displaymath}
\hat{t} = \argmax_{t} \textrm{P}(s|t) \textrm{P}(t)
\end{displaymath}

\begin{itemize}
\item \textbf{������������}
    \begin{enumerate}
    \item \textbf{��ģ}�������������$\textrm{P}(s|t)$��$\textrm{P}(t)$�ļ��㷽ʽ
    \item \textbf{ѵ��}����λ�ü���$\textrm{P}(s|t)$��$\textrm{P}(t)$����Ĳ���
    \item \textbf{����}���������������Ž�Ĺ���$argmax$
    \end{enumerate}
\item&lt;2-&gt; ����һ�±��¿�ʼ��ʵ�����ǲ�����������ʶ�ĸо���
    \vspace{0.5em}
    \begin{center}
    \begin{tikzpicture}
    \node [anchor=west] (e1) at (0,0) {$g(s,t)$};
    \node [anchor=west] (e2) at (e1.east) {$=$};
    \node [anchor=west,inner sep=2pt,fill=red!20] (e3) at (e2.east) {$\prod\nolimits_{(j,i) \in \hat{A}} \textrm{P}(s_j,t_i)$};
    \node [anchor=west,inner sep=1pt] (e4) at (e3.east) {$\times$};
    \node [anchor=west,inner sep=3pt,fill=blue!20] (e5) at (e4.east) {$\textrm{P}_{lm}(t)$};
    \node [anchor=north west,inner sep=1pt] (n1) at ([xshift=2.5em,yshift=-1em]e1.south west) {$\textrm{P}(s|t)$};
    \node [anchor=north] (n1part2) at ([yshift=0.3em]n1.south) {\scriptsize{\textbf{����ģ��}}};
    \node [anchor=west,inner sep=1pt] (n2) at ([xshift=2em]n1.east) {$\textrm{P}(t)$};
    \node [anchor=north] (n2part2) at ([yshift=0.3em]n2.south) {\scriptsize{\textbf{����ģ��}}};
    \draw [-&gt;,thick] (e3.south) .. controls +(south:1em) and +(north:1em) .. (n1.north);
    \draw [-&gt;,thick] (e5.south) .. controls +(south:1em) and +(70:1em) .. (n2.north);
    \end{tikzpicture}
    \end{center}
    \vspace{-0.5em}
    \begin{itemize}
    \item �ʻ㷭����ʺ�ngram����ʹ�����Ƶ�ʹ���({\color{ublue} \hyperref[wordtranslationprob]{��\hspace{-0.2em}~\ref{wordtranslationprob}ҳ}})
    \item $argmax$ʹ��{\color{ublue} \hyperref[simpledecodingalgorithm]{��\hspace{-0.2em}~\ref{simpledecodingalgorithm}ҳ}}�������Ľ����㷨
    \end{itemize}
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{��ģ}

%%%------------------------------------------------------------------------------------------------------------
%%% IBM model-style word alignment
\begin{frame}{IBMģ���еļ��� - �ʶ���}
\label{ibmmodelingstart}

\begin{itemize}
\item \textbf{$\textrm{P}(t)$�ͽ���}��ǰ����������н��ܣ������ص����$\textrm{P}(s|t)$������
    \begin{itemize}
    \item \textbf{����ģ�ͽ�ģ} - $\textrm{P}(s|t)$�ļ��㷽��
    \item \textbf{����ģ�Ͳ�������} - ����$\textrm{P}(s|t)$����Ĳ���
    \end{itemize}
    \vspace{0.5em}
\item&lt;2-&gt; \textbf{IBMģ�͵ļ���}��$s=s_1...s_m$��$t=t_1...t_n$֮���е���һ���Ķ�Ӧ������\alert{���ʶ���}����\alert{�ʶ���}�����⣺
    \begin{itemize}
    \item \textbf{Լ��}��һ��Դ���Ե���ֻ�ܶ�Ӧһ��Ŀ���ﵥ��
    \vspace{0.5em}
    \begin{center}
    \begin{tikzpicture}
    \begin{scope}
    {\small
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=1em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \draw [-] (s1.south) -- ([yshift=-0.2em]t1.north);
    \draw [-] (s2.south) -- (t2.north);
    \node [anchor=center,draw=ublue,circle,thick,fill=white,inner sep=1pt,circular drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (mark) at ([xshift=0.5em,yshift=-0.2em]s2.south east) {{\color{ugreen} \tiny{\textbf{Yes}}}};
    }
    \end{scope}
    \begin{scope}[xshift=1.3in]
    {\small
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=1em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \draw [-] (s1.south) -- ([yshift=-0.2em]t1.north);
    \draw [-] (s1.south) -- (t2.north);
    \node [anchor=center,draw=ublue,circle,thick,fill=white,inner sep=1.5pt,circular drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (mark) at ([xshift=0.5em,yshift=-0.2em]s2.south east) {{\color{red} \tiny{\textbf{No}}}};
    }
    \end{scope}
    \begin{scope}[xshift=2.6in]
    {\small
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=1em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \draw [-] (s1.south) -- ([yshift=-0.2em]t1.north);
    \draw [-] (s2.south) -- ([yshift=-0.2em]t1.north);
    \node [anchor=center,draw=ublue,circle,thick,fill=white,inner sep=1pt,circular drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (mark) at ([xshift=0.5em,yshift=-0.2em]s2.south east) {{\color{ugreen} \tiny{\textbf{Yes}}}};
    }
    \end{scope}
    \end{tikzpicture}
    \end{center}
    \vspace{0.5em}
    \item&lt;3-&gt; \textbf{�շ���(ɾ��)}��Դ���Ե��ʿ��Է��գ���ʱ����Ӧ��һ�������Ŀ���ﵥ��$t_0$
    \vspace{0.3em}
    \begin{center}
    \begin{tikzpicture}
    {\small
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {��};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=1em]s1.east) {����};
    \node [anchor=west,inner sep=2pt] (s3) at ([xshift=1em]s2.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.7em]s1.center) {on};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.5em]s2.center) {the};
    \node [anchor=north,inner sep=2pt] (t3) at ([yshift=-1.5em]s3.center) {table};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-1.5em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t0.north);
    \draw [-] (s2.south) -- (t3.north);
    \draw [-] (s3.south) .. controls +(south:1em) and +(north:1em) .. (t1.north);
    }
    \end{tikzpicture}
    \end{center}
    \end{itemize}
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% modeling P(s|t)
\begin{frame}{��ģ - $\textrm{P}(s|t)$}

\begin{itemize}
\item ����$s$��$t$������֮���\alert{�ʶ���}����Ϊ$a=a_1...a_m$
    \begin{itemize}
    \item $a_j$��ʾ��$j$��Դ�ﵥ��$s_j$��Ӧ��Ŀ���ﵥ�ʵ�λ��
    \begin{center}
    \begin{tikzpicture}
    {\small
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {$s_1$:��};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=1em]s1.east) {$s_2$:����};
    \node [anchor=west,inner sep=2pt] (s3) at ([xshift=1em]s2.east) {$s_3$:��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.7em]s1.center) {$t_1$:on};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.6em]s2.center) {$t_2$:the};
    \node [anchor=north,inner sep=2pt] (t3) at ([yshift=-1.6em]s3.center) {$t_3$:table};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-1.5em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t0.north);
    \draw [-] (s2.south) -- (t3.north);
    \draw [-] (s3.south) -- (t1.north);
    \node [anchor=west] (a) at ([xshift=2em,yshift=-1em]s3.east) {$a_1=0,a_2=3,a_3=1$};
    }
    \end{tikzpicture}
    \end{center}
    \end{itemize}
\item&lt;2-&gt; \textbf{\alert{��ģ!!}}: $\textrm{P}(s|t)$����ʾΪ���п��ܵĴʶ�������ɸ���\\
    \vspace{-0.5em}
    \begin{displaymath}
    \textrm{P}(s|t) = \sum_{a} \textrm{P}(s,a|t)
    \end{displaymath}\\
    \vspace{-0.5em}

    \visible&lt;3-&gt;{
    ÿһ��$a$��Ӧһ��$\textrm{P}(s,a|t)$

    \vspace{-0.8em}
    \begin{center}
    \begin{tikzpicture}
    \begin{scope}
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t0.north);
    \draw [-] (s2.south) -- (t0.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[xshift=1.2in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t0.north);
    \draw [-] (s2.south) -- (t1.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[xshift=2.4in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t0.north);
    \draw [-] (s2.south) -- (t2.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[yshift=-0.5in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- ([yshift=-0.2em]t1.north);
    \draw [-] (s2.south) -- (t0.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[xshift=1.2in,yshift=-0.5in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- ([yshift=-0.2em]t1.north);
    \draw [-] (s2.south) -- ([yshift=-0.2em]t1.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[xshift=2.4in,yshift=-0.5in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- ([yshift=-0.2em]t1.north);
    \draw [-] (s2.south) -- (t2.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[yshift=-1in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t2.north);
    \draw [-] (s2.south) -- (t0.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[xshift=1.2in,yshift=-1in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t2.north);
    \draw [-] (s2.south) -- (t1.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\small{+}};
    }
    }
    \end{scope}
    \begin{scope}[xshift=2.4in,yshift=-1in]
    {\scriptsize
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {лл};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=0.4em]s1.east) {��};
    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-1.2em]s1.center) {thank};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-1.45em]s2.center) {you};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-0.2em,yshift=-0.1em]t1.west) {$t_0$};
    \draw [-] (s1.south) -- (t2.north);
    \draw [-] (s2.south) -- (t2.north);
    \visible&lt;4-&gt;{
    \node [anchor=south east,inner sep=0pt] (p) at (t0.north west) {\small{{\color{ugreen} P(}}};
    \node [anchor=south west,inner sep=0pt] (p2) at ([yshift=0.2em]t2.north east) {\small{{\color{ugreen} )}}};
    \node [anchor=west] (eq) at (p2.east) {\normalsize{= \ P($s|t$)}};
    }
    }
    \end{scope}
    \end{tikzpicture}
    \end{center}
    }

\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% modeling P(s,a|t)
\begin{frame}{��ģ - $\textrm{P}(s,a|t)$}

\begin{itemize}

\item \alert{\textbf{��һ����ģ!!}}������Դ�����$s=s_1...s_m$($m$����)��Ŀ��������$t=t_0...t_n$($n$����)�ʹʶ���$a=a_1...a_m$�������·�ʽ����$\textrm{P}(s,a|t)$

    \begin{itemize}
    \item ���Ŷ��壺$s_x^y=s_x...s_y$, $a_x^y=a_x...a_y$
    \end{itemize}


\vspace{-1.5em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\textrm{P}(s,a|t)=$};
\node [anchor=west,inner sep=1pt,minimum height=2.64em] (eq2) at (eq1.east) {$\textrm{P}(m|t)$};
\node [anchor=west,inner sep=1pt,minimum height=2.64em] (eq3) at ([xshift=1pt]eq2.east) {$\prod\limits_{j=1}^{m}$};
\node [anchor=west,inner sep=1pt,minimum height=2.64em] (eq4) at ([xshift=1pt]eq3.east) {$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t)$};
\node [anchor=west,inner sep=1pt,minimum height=2.64em] (eq5) at ([xshift=1pt]eq4.east) {$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)$};

\visible&lt;2-&gt;{
\node [anchor=west,inner sep=1pt,minimum height=2.64em,fill=red!20] (eq2) at (eq1.east) {$\textrm{P}(m|t)$};
}
\visible&lt;3-&gt;{
\node [anchor=west,inner sep=1pt,minimum height=2.64em,fill=blue!20] (eq3) at ([xshift=1pt]eq2.east) {$\prod\limits_{j=1}^{m}$};
}
\visible&lt;4-&gt;{
\node [anchor=west,inner sep=1pt,minimum height=2.64em,fill=green!20] (eq4) at ([xshift=1pt]eq3.east) {$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t)$};
}
\visible&lt;5-&gt;{
\node [anchor=west,inner sep=1pt,minimum height=2.64em,fill=purple!20] (eq5) at ([xshift=1pt]eq4.east) {$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)$};
}

\visible&lt;2-&gt;{
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c1) at ([yshift=-1pt]eq2.south) {\tiny{\textbf{1}}};
}
\visible&lt;3-&gt;{
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c2) at ([yshift=-1pt]eq3.south) {\tiny{\textbf{2}}};
}
\visible&lt;4-&gt;{
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c3) at ([yshift=-1pt]eq4.south) {\tiny{\textbf{3}}};
}
\visible&lt;5-&gt;{
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c4) at ([yshift=-1pt]eq5.south) {\tiny{\textbf{4}}};
}

\end{tikzpicture}
\end{center}

\vspace{-0.0em}

    \begin{itemize}
    \item&lt;2-&gt; \textbf{����ģ��}����������$t$����Դ��$s$�Ͷ���$a$
        \begin{enumerate}
        \item&lt;2-&gt; ��������$t$ѡ��Դ�ĵĳ���$m$
        \item&lt;3-&gt; ѭ��Դ�ĵ�ÿ��λ��$j$
        \item&lt;4-&gt; ��������$t$��Դ�ĳ���$m$���Ѿ����ɵ�Դ�ﵥ��$s_{1}^{j-1}$�Ͷ���$a_{1}^{j-1}$�����ɵ�$j$��λ�õĶ�����$a_j$
        \item&lt;5-&gt; ��������$t$��Դ�ĳ���$m$���Ѿ����ɵ�Դ�ﵥ��$s_{1}^{j-1}$�Ͷ���$a_{1}^{j}$�����ɵ�$j$��λ�õ�Դ���Ե���$s_j$��ע�⣺��ʱ$a_j$�Ѿ������ˣ�
        \end{enumerate}
    \end{itemize}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% generation of s and a given t -  a running example
\begin{frame}{ʵ�� - $\textrm{P}(s,a|t)$}

$s$ = �� ���� �� \ \ \ \ \ $t$ = $t_0$ on the table \ \ \ \ \ $a$ = \{1-0,2-3,3-1\}

\begin{center}
\begin{tikzpicture}
\visible&lt;1,5-&gt;{
\node [anchor=west,inner sep=2pt] (s1) at (0,0) {$s_1$:��};
}
\visible&lt;1,7-&gt;{
\node [anchor=west,inner sep=2pt] (s2) at ([xshift=2em]s1.east) {$s_2$:����};
}
\visible&lt;1&gt;{
\node [anchor=west,inner sep=2pt] (s3) at ([xshift=2em]s2.east) {$s_3$:��};
}
\node [anchor=north,inner sep=2pt] (t1) at ([yshift=-2.4em]s1.center) {$t_1$:on};
\node [anchor=north,inner sep=2pt] (t2) at ([yshift=-2.4em]s2.center) {$t_2$:the};
\node [anchor=north,inner sep=2pt] (t3) at ([yshift=-2.4em]s3.center) {$t_3$:table};
\node [anchor=east,inner sep=2pt] (t0) at ([xshift=-2.2em]t1.west) {$t_0$};

\visible&lt;1,4-&gt;{
\draw [-,thick] (s1.south) -- (t0.north);
}
\visible&lt;1,6-&gt;{
\draw [-,thick] (s2.south) -- (t3.north);
}
\visible&lt;1,8-&gt;{
\draw [-,thick] (s3.south) .. controls +(south:1.5em) and +(north:1.5em) .. (t1.north);
}

\visible&lt;2-3&gt;{
\node [anchor=center] at (s1) {$\bullet$};
}

\visible&lt;2-5&gt;{
\node [anchor=center] at (s2) {$\bullet$};
}

\visible&lt;2-7&gt;{
\node [anchor=center] at (s3) {$\bullet$};
}

\visible&lt;4&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20] (s1) at (0,0) {$s_1$:��};
}
\visible&lt;6&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20] (s2) at ([xshift=2em]s1.east) {$s_2$:����};
}
\visible&lt;8&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20] (s3) at ([xshift=2em]s2.east) {$s_3$:��};
}
\visible&lt;3&gt;{
\draw [-,thick,red] (s1.south) -- (t0.north);
}
\visible&lt;5&gt;{
\draw [-,thick,red] (s2.south) -- (t3.north);
}
\visible&lt;7&gt;{
\draw [-,thick,red] (s3.south) .. controls +(south:1.5em) and +(north:1.5em) .. (t1.north);
}

\end{tikzpicture}
\end{center}

\vspace{-1.5em}

{\small
\begin{eqnarray}
\textrm{P}(s,a|t) &amp; = &amp; \textrm{P}(m|t) \prod\limits_{j=1}^{m} \textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t) \textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t) \nonumber \\
                  &amp; \visible&lt;2-&gt;{=} &amp; \visible&lt;2-&gt;{\textrm{P}(m=3 \mid \textrm{'$t_0$ on the table'})} \visible&lt;3-&gt;{\times} \nonumber \\
                  &amp;   &amp; \visible&lt;3-&gt;{\textrm{P}(a_1=0 \mid \phi,\phi,3,\textrm{'$t_0$ on the table'})} \visible&lt;4-&gt;{\times} \nonumber \\
                  &amp;   &amp; \visible&lt;4-&gt;{\textrm{P}(f_1=\textrm{��} \mid \textrm{\{1-0\}},\phi,3,\textrm{'$t_0$ on the table'})} \visible&lt;5-&gt;{\times} \nonumber \\
                  &amp;   &amp; \visible&lt;5-&gt;{\textrm{P}(a_2=3 \mid \textrm{\{1-0\}},\textrm{'��'},3,\textrm{'$t_0$ on the table'})} \visible&lt;6-&gt;{\times} \nonumber \\
                  &amp;   &amp; \visible&lt;6-&gt;{\textrm{P}(f_2=\textrm{����} \mid \textrm{\{1-0,2-3\}},\textrm{'��'},3,\textrm{'$t_0$ on the table'})} \visible&lt;7-&gt;{\times} \nonumber \\
                  &amp;   &amp; \visible&lt;7-&gt;{\textrm{P}(a_3=1 \mid \textrm{\{1-0,2-3\}},\textrm{'�� ����'},3,\textrm{'$t_0$ on the table'})} \visible&lt;8-&gt;{\times} \nonumber \\
                  &amp;   &amp; \visible&lt;8-&gt;{\textrm{P}(f_3=\textrm{��} \mid \textrm{\{1-0,2-3,3-1\}},\textrm{'�� ����'},3,\textrm{'$t_0$ on the table'})} \nonumber
\end{eqnarray}
}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% problems
\begin{frame}{��������}

\begin{itemize}

\item \textbf{\alert{�ص�}}�����»ع�һ�����ģ�� - ������ʽ

\vspace{-1.0em}

\begin{eqnarray}
\textrm{P}(s|t) &amp; = &amp; \sum_{a} \textrm{P}(s,a|t)  \nonumber \\
\textrm{P}(s,a|t) &amp; = &amp; \textrm{P}(m|t) \prod\limits_{j=1}^{m} \textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t) \textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)  \nonumber
\end{eqnarray}

\item \textbf{������������}
    \begin{enumerate}
    \item ��һ����ʽ����α������еĶ���$a$
    \item �ڶ�����ʽ����μ���$\textrm{P}(m|t)$��$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t)$��$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)$
    \end{enumerate}

\item&lt;2-&gt; Brown����(1993)�Ľ����������������л���
    \begin{itemize}
    \item IBM Models 1-2: ����ʽ2�Ĳ�������ȷ��Ч��⹫ʽ1\\
    \alert{$\to$���̳�����}
    \item IBM Models 3-5: �򵥻���ʽ2�Ĳ�����������⹫ʽ1\\
    \alert{$\to$��ѧ����}
    \end{itemize}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{IBMģ��1}

%%%------------------------------------------------------------------------------------------------------------
%%% assumptions in IBM Model 1
\begin{frame}{IBMģ��1�ļ���}

\begin{itemize}

\item \textbf{IBMģ��1����}
    \begin{enumerate}
    \item Դ�ﳤ�ȸ���Ϊ����$\epsilon$
    \begin{displaymath}
    \textrm{P}(m|t) \equiv \epsilon
    \end{displaymath}
    \item �������$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t)$�����������ĳ���$l+1$(���ȷֲ�)
    \begin{displaymath}
    \textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t) \equiv \frac{1}{l+1}
    \end{displaymath}
    \item Դ�ﵥ��$s_j$���ɸ���$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)$�����������������ĵ���$t_{a_j}$�����ʻ㷭�����$f(s_j|t_{a_j})$ ($\sum_{s_j} f(s_j|t_{a_j}) = 1$)
    \begin{displaymath}
    \textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t) \equiv f(s_j|t_{a_j})
    \end{displaymath}
    \end{enumerate}

\item&lt;2-&gt; \textbf{����˼����}�Ѹ��Ӳ�������Ϊ�򵥲���
    \begin{itemize}
    \item ���磺$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t) \equiv \frac{1}{l-1}$�Ѳ����ռ�$(a_{1}^{j},s_{1}^{j-1},m,t)$����Ϊ$l$
    \item \alert{�ŵ�}: ģ�ʹ�󻯼�\alert{ȱ��}��������ģ�Ͳ�׼ȷ
    \end{itemize}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% the full model of IBM Model 1
\begin{frame}{IBMģ��1 - ����ģ��}

\begin{itemize}
\item \textbf{�������}

\vspace{-1em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\textrm{P}(s,a|t)$};
\node [anchor=west] (eq1part2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=1pt] (eq2) at (eq1part2.east) {$\textrm{P}(m|t)$};
\node [anchor=west,inner sep=1pt] (eq3) at ([xshift=1pt]eq2.east) {$\prod\limits_{j=1}^{m}$};
\node [anchor=west,inner sep=1pt] (eq4) at ([xshift=1pt]eq3.east) {$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t)$};
\node [anchor=west,inner sep=1pt] (eq5) at ([xshift=1pt]eq4.east) {$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)$};

\node [anchor=east,rotate=90] (yes1) at (eq2.south) {$\equiv$};
\node [anchor=east,rotate=90] (yes2) at (eq4.south) {$\equiv$};
\node [anchor=east,rotate=90] (yes3) at (eq5.south) {$\equiv$};

\node [anchor=north,inner sep=1pt] at (yes1.west) {$\epsilon$};
\node [anchor=north,inner sep=1pt] at (yes2.west) {$\frac{1}{l+1}$};
\node [anchor=north,inner sep=1pt] at (yes3.west) {$f(s_j|t_{a_j})$};

\node [anchor=west] (neweq1) at ([yshift=-5em]eq1part2.west) {$=$};
\node [anchor=west,inner sep=1pt] (neweq2) at (neweq1.east) {$\epsilon \prod\limits_{j=1}^{m} \frac{1}{l+1} f(s_j|t_{a_j})$};

\node [anchor=west] (new2eq1) at ([yshift=-2.5em]neweq1.west) {$=$};
\node [anchor=west,inner sep=1pt] (new2eq2) at (new2eq1.east) {$\frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} f(s_j|t_{a_j})$};

\end{tikzpicture}
\end{center}

\item&lt;2-&gt; \textbf{����ʽ����$\textrm{P}(s|t)=\sum_a \textrm{P}(s,a|t)$}

\vspace{-1.0em}

\begin{eqnarray}
\textrm{P}(s|t) &amp; = &amp; \sum\limits_{a} \textrm{P}(s,a|t) \nonumber \\
                &amp; = &amp; \sum\limits_{a} \frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} f(s_j|t_{a_j}) \nonumber
\end{eqnarray}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% full model of IBM Model 1 - cont.
\begin{frame}{IBMģ��1 - ����ģ��(2)}

\begin{itemize}

\item $\textrm{P}(s|t) = \sum\limits_{a} \frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} f(s_j|t_{a_j})$����Ҫ�Ա������еĶ��룬��$\sum_{a}$��������̿��Ա����±�ʾΪ

\vspace{-0.5em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\textrm{P}(s|t)$};
\node [anchor=west] (eq2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq3) at (eq2.east) {$\sum\limits_{a_1=0}^{l}$};
\node [anchor=west,inner sep=0] (eq4) at (eq3.east) {...};
\node [anchor=west,inner sep=2pt] (eq5) at (eq4.east) {$\sum\limits_{a_m=0}^{l}$};
\node [anchor=west,inner sep=2pt] (eq6) at ([xshift=0.2em]eq5.east) {$\frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} f(s_j|t_{a_j})$};

\visible&lt;3&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20] (eq3) at (eq2.east) {$\sum\limits_{a_1=0}^{l}$};
}
\visible&lt;4&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20] (eq5) at (eq4.east) {$\sum\limits_{a_m=0}^{l}$};
}

\visible&lt;6-&gt;{
\node [anchor=east] (label) at ([xshift=-0.5em,yshift=0.1em]eq1.west) {\alert{\textbf{IBMģ��1:}}};
}

\begin{pgfonlayer}{background}
\visible&lt;2-5&gt;{
\node[rectangle,draw,ublue,inner sep=0.1em,fill=white] [fit = (eq3) (eq5)] (align) {};
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c1) at ([yshift=-1pt]align.south) {\tiny{\textbf{1}}};
}
\visible&lt;5&gt;{
\node[rectangle,draw,ublue,inner sep=0.0em,minimum height=3.2em,fill=white] [fit = (eq6)] (prob) {};
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c1) at ([yshift=-1pt]prob.south) {\tiny{\textbf{2}}};
}
\visible&lt;6-&gt;{
\node[rectangle,draw,thick,red,inner sep=0.3em,minimum height=3em,fill=white] [fit = (label) (eq6)] (prob) {};
}
\end{pgfonlayer}

\end{tikzpicture}
\end{center}

    \begin{enumerate}
    \item&lt;2-&gt; �������еĵĶ���$a$��$a$��\{$a_1$,...,$a_m$\}��ɣ�ÿ��$a_j \in \{a_1,...,a_m\}$�ӵ����Ŀ�ʼλ��(0)ѭ������ֹλ��($l$)

    \vspace{0.5em}
    \begin{center}
    \begin{tikzpicture}

    \visible&lt;3-&gt;{
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {$s_1$:��};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=2em]s1.east) {$s_2$:����};
    \node [anchor=west,inner sep=2pt] (s3) at ([xshift=2em]s2.east) {$s_3$:��};

    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-2.4em]s1.center) {$t_1$:on};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-2.4em]s2.center) {$t_2$:the};
    \node [anchor=north,inner sep=2pt] (t3) at ([yshift=-2.4em]s3.center) {$t_3$:table};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-2.2em]t1.west) {$t_0$};
    }

    \visible&lt;3&gt;{
    \draw [-,dashed,thick] (s1.south) -- (t0);
    \draw [-,dashed,thick] (s1.south) -- (t1);
    \draw [-,dashed,thick] (s1.south) -- (t2);
    \draw [-,dashed,thick] (s1.south) -- (t3);
    }

    \visible&lt;4-&gt;{
    \draw [-,dashed,thick] (s3.south) -- (t0);
    \draw [-,dashed,thick] (s3.south) -- (t1);
    \draw [-,dashed,thick] (s3.south) -- (t2);
    \draw [-,dashed,thick] (s3.south) -- (t3);
    }

    \end{tikzpicture}
    \end{center}
    \vspace{0.5em}

    \item&lt;5-&gt; ����ÿ��$a$�ۼӶ������$\textrm{P}(s,a|t)$
    \end{enumerate}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{IBMģ��2}

%%%------------------------------------------------------------------------------------------------------------
%%% the assumptions in IBM Model 2
\begin{frame}{IBMģ��2�ļ���}

\begin{itemize}

\item IBMģ��1��Ȼ��ʽ�ǳ��򵥣����������Ǻ����˷���ĵ������⡣����ͬһ��Դ���Ծ��ӣ����ʷ�����ͬ����������˳��ͬ��ģ�͵÷֣����������ͬ��

\begin{center}
\begin{tikzpicture}

\begin{scope}

\node [anchor=west] (s1) at (0,0) {$s$ = ��\ \ ����\ \ ��};
\node [anchor=west] (t1) at ([yshift=-2em]s1.west) {$t$ = on\ \ the\ \ table};
\draw [-&gt;,double,thick,ublue] ([yshift=0.2em]s1.south) -- ([yshift=-0.8em]s1.south);

\end{scope}

\begin{scope}[xshift=1.5in]

\node [anchor=west] (s2) at (0,0) {$s$ = ��\ \ ����\ \ ��};
\node [anchor=west] (t2) at ([yshift=-2em]s2.west) {$t'$ = table \ on\ \ the};
\draw [-&gt;,double,thick,ublue] ([yshift=0.2em]s2.south) -- ([yshift=-0.8em]s2.south);

\end{scope}

\node [anchor=north] (score11) at ([yshift=-2.0em]s1.south) {$\textrm{P}(s|t)$};
\node [anchor=north] (score12) at ([yshift=-2.0em]s2.south) {$\textrm{P}(s|t')$};
\node [anchor=west] (comp1) at ([xshift=2.3em]score11.east) {\large{$\mathbf{=}$}};
\node [anchor=east] (label1) at ([xshift=-1em,yshift=0.1em]score11.west) {\textbf{IBMģ��1:}};

\visible&lt;2-&gt;{
\node [anchor=north] (score21) at ([yshift=0.2em]score11.south) {$\textrm{P}(s|t)$};
\node [anchor=north] (score22) at ([yshift=0.2em]score12.south) {$\textrm{P}(s|t')$};
\node [anchor=west] (comp2) at ([xshift=2.3em]score21.east) {\large{$\mathbf{&gt;}$}};
\node [anchor=east] (label2) at ([xshift=-1em,yshift=0.1em]score21.west) {\textbf{����:}};
}

\end{tikzpicture}
\end{center}

\vspace{-0.5em}

\item&lt;3-&gt; \textbf{IBMģ��2�ļ���}������λ��$a_j$�����ɸ��������Ե���λ��$j$��Դ����ӳ���$m$�����ĳ��Ⱥ�$l$�йء���Ϊ$a(a_j|j,m,l)$������$a_j$��\alert{�������}
    \vspace{-0.8em}

    \begin{displaymath}
    \textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t) \equiv a(a_j|j,m,l)
    \end{displaymath}

    ����������IBMģ��1��ͬ����$\textrm{P}(m|t) \equiv \epsilon$��$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t) \equiv f(s_j|t_{a_j})$

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% the full model in IBM Model 2
\begin{frame}{IBMģ��2 - ����ģ��}

\begin{itemize}

\item \textbf{�������}

\vspace{-1em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\textrm{P}(s,a|t)$};
\node [anchor=west] (eq1part2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=1pt] (eq2) at (eq1part2.east) {$\textrm{P}(m|t)$};
\node [anchor=west,inner sep=1pt] (eq3) at ([xshift=1pt]eq2.east) {$\prod\limits_{j=1}^{m}$};
\node [anchor=west,inner sep=1pt] (eq4) at ([xshift=1pt]eq3.east) {$\textrm{P}(a_j|a_{1}^{j-1},s_{1}^{j-1},m,t)$};
\node [anchor=west,inner sep=1pt] (eq5) at ([xshift=1pt]eq4.east) {$\textrm{P}(s_j|a_{1}^{j},s_{1}^{j-1},m,t)$};

\node [anchor=east,rotate=90] (yes1) at (eq2.south) {$\equiv$};
\node [anchor=east,rotate=90] (yes2) at (eq4.south) {$\equiv$};
\node [anchor=east,rotate=90] (yes3) at (eq5.south) {$\equiv$};

\node [anchor=north,inner sep=1pt] at (yes1.west) {$\epsilon$};
\node [anchor=north,inner sep=1pt] at (yes2.west) {$a(a_j|j,m,l)$};
\node [anchor=north,inner sep=1pt] at (yes3.west) {$f(s_j|t_{a_j})$};

\node [anchor=west] (neweq1) at ([yshift=-5em]eq1part2.west) {$=$};
\node [anchor=west,inner sep=1pt] (neweq2) at (neweq1.east) {$\epsilon \prod\limits_{j=1}^{m} a(a_j|j,m,l) f(s_j|t_{a_j})$};

\end{tikzpicture}
\end{center}

\item&lt;2-&gt; \textbf{����ʽ����$\textrm{P}(s|t)=\sum_a \textrm{P}(s,a|t)$}

\vspace{-0.5em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\textrm{P}(s|t)$};
\node [anchor=west] (eq2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq3) at (eq2.east) {$\sum\limits_{a_1=0}^{l}$};
\node [anchor=west,inner sep=0] (eq4) at ([xshift=-0.2em]eq3.east) {...};
\node [anchor=west,inner sep=2pt] (eq5) at ([xshift=-0.4em]eq4.east) {$\sum\limits_{a_m=0}^{l}$};
\node [anchor=west,inner sep=2pt] (eq6) at ([xshift=0.2em]eq5.east) {$\epsilon \prod\limits_{j=1}^{m} a(a_j|j,m,l) f(s_j|t_{a_j})$};

\visible&lt;3-&gt;{
\node [anchor=east] (label) at ([xshift=-0.0em,yshift=0.1em]eq1.west) {\alert{\textbf{IBMģ��2:}}};
}

\begin{pgfonlayer}{background}
\visible&lt;2&gt;{
\node[rectangle,draw,ublue,inner sep=0.1em,fill=white] [fit = (eq3) (eq5)] (align) {};
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c1) at ([yshift=-1pt]align.south) {\tiny{\textbf{1}}};
}
\visible&lt;2&gt;{
\node[rectangle,draw,ublue,inner sep=0.0em,minimum height=3.2em,fill=white] [fit = (eq6)] (prob) {};
\node [anchor=north,draw,circle,inner sep=1pt,ublue] (c1) at ([yshift=-1pt]prob.south) {\tiny{\textbf{2}}};
}
\visible&lt;3-&gt;{
\node[rectangle,draw,thick,red,inner sep=0.3em,minimum height=3em,fill=white] [fit = (label) (eq6)] (prob) {};
}
\end{pgfonlayer}

\end{tikzpicture}
\end{center}

\vspace{-0.5em}

    \begin{enumerate}
    \item �������еĵĶ���$a$
    \item ����ÿ��$a$�ۼӶ������$\textrm{P}(s,a|t)$��������$\epsilon \prod\limits_{j=1}^{m} a(a_j|j,m,l) f(s_j|t_{a_j})$
    \end{enumerate}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% review the IBM Models 1-2
%%% model parameters
\begin{frame}{ͣ��һ�£����¿���������ģ��}

\vspace{-1.0em}

\begin{eqnarray}
\textrm{\textbf{IBMģ��1}}: \textrm{P}(s|t) &amp; = &amp; \frac{\epsilon}{(l+1)^{m}} \sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m} f(s_j|t_{a_j}) \nonumber \\
\textrm{\textbf{IBMģ��2}}: \textrm{P}(s|t) &amp; = &amp; \epsilon \sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m} a(a_j|j,m,l) f(s_j|t_{a_j}) \nonumber
\end{eqnarray}

\begin{itemize}

\item ����
    \begin{itemize}
    \item $\epsilon$Ϊ����
    \item&lt;2-&gt; $f(s_j|t_{a_j})$: ��������$t_{a_j}$����Ϊ$s_j$�ĸ���
    \item&lt;3-&gt; $a(a_j|j,m,l)$: ����λ��$j$�;�Գ���$m$��$l$������$a_j$�ĸ���
    \end{itemize}

    \begin{center}
    \begin{tikzpicture}

    \visible&lt;2-&gt;{
    \node [anchor=west,inner sep=2pt] (s1) at (0,0) {$s_1$:��};
    \node [anchor=west,inner sep=2pt] (s2) at ([xshift=2em]s1.east) {$s_2$:����};
    \node [anchor=west,inner sep=2pt] (s3) at ([xshift=2em]s2.east) {$s_3$:��};

    \node [anchor=north,inner sep=2pt] (t1) at ([yshift=-2.4em]s1.center) {$t_1$:on};
    \node [anchor=north,inner sep=2pt] (t2) at ([yshift=-2.4em]s2.center) {$t_2$:the};
    \node [anchor=north,inner sep=2pt] (t3) at ([yshift=-2.4em]s3.center) {$t_3$:table};
    \node [anchor=east,inner sep=2pt] (t0) at ([xshift=-2.2em]t1.west) {$t_0$};

    \draw [-,dashed,thick] (s1.south) -- (t0.north);
    \draw [-,dashed,thick] (s2.south) -- (t3.north);
    \draw [-,dashed,thick] (s3.south) -- (t1.north);
    }

    \begin{pgfonlayer}{background}
    \visible&lt;2&gt;{
    \path [fill=red!20] (s2.north west) -- (s2.south west) -- (t3.north west) -- (t3.south west) -- (t3.south east) -- (t3.north east) -- (s2.south east) -- (s2.north east) -- (s2.north west);
    }
    \end{pgfonlayer}

    \visible&lt;2&gt;{
    \node [anchor=south,inner sep=1pt] (mark1) at (s2.north) {\alert{\tiny{��ʾ����'table'����Ϊ'����'�ĸ���}}};
    \node [anchor=south west,inner sep=1pt] (mark1part2) at (mark1.north west) {\alert{\tiny{$f(s_2|t_{a_2})=f(\textrm{����}|\textrm{table})$}}};
    }

    \visible&lt;3-&gt;{
    \draw [-,dashed,thick,red] (s2.south) -- (t3.north);

    \node [anchor=west,inner sep=1pt] (mark2) at ([xshift=0.1in]s3.south east) {\alert{\tiny{$a(a_j|j,m,l)=a(3|2,3,3)$}}};
    \node [anchor=north west,inner sep=1pt] (mark2part2) at (mark2.south west) {\alert{\tiny{��ʾ����λ��$j=2$}}};
    \node [anchor=north west,inner sep=1pt] (mark2part3) at (mark2part2.south west) {\alert{\tiny{���ӳ���$m=3$��$l=3$}}};
    \node [anchor=north west,inner sep=1pt] (mark2part4) at (mark2part3.south west) {\alert{\tiny{���뵽Ŀ����λ��3($a_j=3$)�ĸ���}}};

    \draw [-&gt;] (mark2.south west) -- ([yshift=0.1in,xshift=-0.1in]t3.north);
    }

    \end{tikzpicture}
    \end{center}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% review the IBM Models 1-2
%%% two problems - the computation problem and the training problem
\begin{frame}{ͣ��һ�£����¿���������ģ��(2)}
\label{ibmmodelingend}

\vspace{-1.0em}

\begin{eqnarray}
\textrm{\textbf{IBMģ��1}}: \textrm{P}(s|t) &amp; = &amp; \frac{\epsilon}{(l+1)^{m}} \sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m} f(s_j|t_{a_j}) \nonumber \\
\textrm{\textbf{IBMģ��2}}: \textrm{P}(s|t) &amp; = &amp; \epsilon \sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m} a(a_j|j,m,l) f(s_j|t_{a_j}) \nonumber
\end{eqnarray}

\begin{spacing}{1.2}
\begin{itemize}

\item ���ڷ���ģ��$\textrm{P}(s|t)$�������ع�һ��ͳ�ƻ��������������������
    \begin{enumerate}
    \item \textbf{��ģ}���������$\textrm{P}(s|t)$ \visible&lt;2-&gt;{\alert{$\gets$ �ѽ⣡������������ʽ}}
    \item \textbf{����}������ģ�Ͳ���$\epsilon$��$a(a_j|j,m,l)$��$f(s_j|t_{a_j})$�������������Ĺ�ʽ����$\textrm{P}(s|t)$(����ģ�ͼ����ݲ�����)�����ҵ��������$\hat{t}$ \visible&lt;2-&gt;{\alert{$\gets$ ��������}}
    \item \textbf{ѵ��}����δ��������Զ�ѧϰģ�Ͳ���$\epsilon$��$a(a_j|j,m,l)$��$f(s_j|t_{a_j})$ \visible&lt;2-&gt;{\alert{$\gets$ ��������}}
    \end{enumerate}

\end{itemize}
\end{spacing}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{����}

%%%------------------------------------------------------------------------------------------------------------
%%% outline
\begin{frame}{Outline}

\begin{center}
\begin{tikzpicture}

{\Large
\node [anchor=west,draw,gray,very thick,rounded corners=4pt,text width=15em,align=left,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (p1) at (0,0) {\gray{\textbf{1. һ���򵥵ķ���ʵ��}}};

\node [anchor=north west,draw,ublue,very thick,rounded corners=4pt,text width=15em,align=left,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (p21) at ([yshift=-1em]p1.south west) {\black{\textbf{2. IBMģ��}}\\\gray{\textbf{\hspace{1.5em}$\bullet$  ��ģ}}\\\black{\textbf{\hspace{1.5em}$\bullet$  ����}}\\\gray{\textbf{\hspace{1.5em}$\bullet$  ģ��ѵ��}}};
}

\end{tikzpicture}
\end{center}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% decoding with IBM Models 1-2
\begin{frame}{���� = ���� + ģ�͵÷ּ���}
\label{ibmdecoding}

\begin{itemize}

\item \textbf{��Ϊģ��}��ͳ������ + ����

\vspace{-0.8em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (model) at (0,0) {\footnotesize{$\textrm{P}(s|t) = \epsilon \sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m} a(a_j|j,m,l) f(s_j|t_{a_j})$}};
\node [anchor=east] (modellabel) at ([yshift=0.1em]model.west) {\footnotesize{\textbf{�����ͳ������:}}};
\node [anchor=north west] (paras) at (model.south west) {\footnotesize{$\epsilon = ?;\ \ \forall a_j,j,m,l: a(a_j|j,m,l) = ?, f(s_j|t_{a_j}) = ?$}};
\node [anchor=east] (paraslabel) at ([yshift=0.1em]paras.west) {\footnotesize{\textbf{ģ�͵Ĳ���:}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,red,inner sep=0.1em,fill=white] [fit = (model) (modellabel) (paras) (paraslabel)] {};
\end{pgfonlayer}

\end{tikzpicture}
\end{center}

\item&lt;2-&gt; \textbf{��Ϊ����}��ģ�͵÷ּ��� + �ҵ�ģ�͵÷���ߵ�����

\vspace{-0.8em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (scoring) at (0,0) {\footnotesize{�������$s$��$t$��(��Ч��)����$\textrm{P}(s|t)$(ͬʱ����$\textrm{P}(t)$)}};
\node [anchor=east] (scoringlabel) at ([yshift=0.1em]scoring.west) {\footnotesize{\textbf{ģ�͵÷ּ���:}}};
\node [anchor=north west] (search) at (scoring.south west) {\footnotesize{�����п��ܵ�$t$���ҵ�ģ�͵÷�($\textrm{P}(s|t)\textrm{P}(t)$)���}};
\node [anchor=north west] (searchpart2) at ([yshift=0.3em]search.south west) {\footnotesize{���������}};
\node [anchor=east] (searchlabel) at ([yshift=0.1em]search.west) {\footnotesize{\textbf{����:}}};

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,red,inner sep=0.1em,fill=white] [fit = (scoring) (scoringlabel) (search) (searchlabel) (searchpart2)] {};
\end{pgfonlayer}

\end{tikzpicture}
\end{center}

\vspace{-0.7em}

    \begin{itemize}
    \item \textbf{����(����)����}ǰ���ʵ���Ѿ�������һ�ֽⷨ����{\color{ublue} \hyperref[simpledecodingalgorithm]{��\hspace{-0.2em}~\ref{simpledecodingalgorithm}ҳ}}�������������������ĵ��� + ��֦���������ﲻ�����ۣ���������ѧϰ
    \item \textbf{ʣ�µ�������}�����������$s$��$t$�����\alert{��Ч��}����$\textrm{P}(s|t)$
    \end{itemize}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% efficient ways of computing model score
\begin{frame}{ģ�͵÷ֵļ���}

\begin{itemize}

\item $O((l+1)^m \cdot m)$ - IBMģ�͵÷ֵ�ֱ�Ӽ��㼸�������ܣ�

\begin{displaymath}
\textrm{P}(s|t) = \frac{\epsilon}{(l+1)^{m}} \underbrace{\sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l}}_{(l+1)^m\textrm{��ѭ��}} \underbrace{\prod\limits_{j=1}^{m} f(s_j|t_{a_j})}_{m\textrm{��ѭ��}}
\end{displaymath}

\item&lt;2-&gt; $O(l \cdot m)$ - ʵ�������ǿ������ĸ���

\vspace{-0.5em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m} f(s_j|t_{a_j})$};
\node [anchor=west] (eq1part2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq1part3) at (eq1part2.east) {$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$};
\visible&lt;3-&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20] (eq1part3) at (eq1part2.east) {$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$};
}

\visible&lt;2&gt;{
\draw[decorate,thick,decoration={brace,amplitude=5pt,mirror}] ([xshift=0.2em,yshift=0.2em]eq1part3.south west) -- ([xshift=-0.2em,yshift=0.2em]eq1part3.south east) node [pos=0.5,below,yshift=-0.2em] (label2) {\scriptsize{\textbf{$m \cdot l$��ѭ��}}};
}

\visible&lt;3-&gt;{
\node [anchor=west] (eq2) at ([xshift=5em,yshift=-4.5em]eq1.west) {$\textrm{P}(s|t) = \frac{\epsilon}{(l+1)^{m}} $};
\node [anchor=west,inner sep=2pt] (eq2part2) at ([xshift=-0.3em]eq2.east) {$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$};
\node [anchor=east] (eq2label) at ([xshift=-0em,yshift=0.2em]eq2.west) {\small{IBMģ��1:}};

\node [anchor=west,inner sep=2pt,fill=red!20] (eq2part2) at ([xshift=-0.3em]eq2.east) {$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$};
}

\visible&lt;4-&gt;{
\node [anchor=west] (eq3) at ([xshift=5em,yshift=-7.5em]eq1.west) {$\textrm{P}(s|t) = \epsilon$};
\node [anchor=west,inner sep=2pt] (eq3part2) at ([xshift=-0.3em]eq3.east) {$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} a(i|j,m,l) f(s_j|t_i)$};
\node [anchor=east] (eq3label) at ([xshift=-0em,yshift=0.2em]eq3.west) {\small{���Ƶģ�IBMģ��2:}};
}

\begin{pgfonlayer}{background}
\visible&lt;4-&gt;{
\node[rectangle,draw=red,inner sep=2pt] [fit = (eq2) (eq2part2) (eq2label) (eq3) (eq3part2) (eq3label)] {};
}
\end{pgfonlayer}

\visible&lt;3-&gt;{
\draw [-&gt;,thick] ([xshift=-1em]eq1part3.south) .. controls +(south:1.3em) and +(north:1.3em) .. ([xshift=1em]eq2part2.north);
}

\end{tikzpicture}
\end{center}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% the trick used in computing the score
\begin{frame}{$\sum_{a_1=0}^{l} ... \sum_{a_m=0}^{l} \prod_{j=1}^{m} f(s_j|t_{a_j})=\prod_{j=1}^{m} \sum_{i=0}^{l} f(s_j|t_i)$}

\begin{itemize}

\item ����$\alpha(x,y)$Ϊһ��������$x$��$y$����������

\end{itemize}

\vspace{-0.0em}

\begin{center}
\begin{tikzpicture}

{\footnotesize
\node [anchor=west] (mid) at (0,0) {$\alpha(1,0)\alpha(2,0) + \alpha(1,0)\alpha(2,1) + \alpha(1,0)\alpha(2,2) +$};
\node [anchor=west] (mid2) at ([yshift=-2em]mid.west) {$\alpha(1,1)\alpha(2,0) + \alpha(1,1)\alpha(2,1) + \alpha(1,1)\alpha(2,2)+$};
\node [anchor=west] (mid3) at ([yshift=-2em]mid2.west) {$\alpha(1,2)\alpha(2,0) + \alpha(1,2)\alpha(2,1) + \alpha(1,2)\alpha(2,2)$};
}

\begin{pgfonlayer}{background}
\node[rectangle,draw=ublue,red,inner sep=0.1em,fill=white] [fit = (mid) (mid2) (mid3)] (exampleeq) {};
\end{pgfonlayer}

{\footnotesize
\visible&lt;2-&gt;{
\node [anchor=north] (eq1) at ([xshift=2em,yshift=-2em]exampleeq.south west) {$\sum\limits_{y_1=0}^{2} \sum\limits_{y_2=0}^{2} \alpha(1,y_1)\alpha(2,y_2)$};
\node [anchor=west] (eq1part2) at ([xshift=-1em,yshift=-3em]eq1.west) {$=$};
\node [anchor=west] (eq1part3) at ([xshift=-0.5em]eq1part2.east) {$\sum\limits_{y_1=0}^{2} \sum\limits_{y_2=0}^{2} \prod\limits_{x=1}^{2} $};
\node [anchor=west,inner sep=2pt] (eq1part4) at ([xshift=-0.3em]eq1part3.east) {$\alpha(x,y_x)$};
}

\visible&lt;3-&gt;{
\node [anchor=north] (eq2) at ([xshift=-2em,yshift=-2em]exampleeq.south east) {$(\alpha(1,0)+\alpha(1,1)+\alpha(1,2))\cdot$};
\node [anchor=west] (eq2part2) at ([yshift=-1.5em]eq2.west) {$(\alpha(2,0)+\alpha(2,1)+\alpha(2,2))$};
\node [anchor=west] (eq2part3) at ([xshift=2.1in]eq1part2.east){$=$};
\node [anchor=west] (eq2part4) at ([xshift=-0.5em]eq2part3.east){$\prod\limits_{x=1}^{2} \sum\limits_{y=0}^{2}$};
\node [anchor=west,inner sep=2pt] (eq2part5) at ([xshift=-0.3em]eq2part4.east){$\alpha(x,y)$};
}
}

\begin{pgfonlayer}{background}
\visible&lt;2-&gt;{
\node[rectangle,draw=ublue,red,inner sep=0.1em,fill=white] [fit = (eq1) (eq1part2) (eq1part3)] (eq1full) {};
}
\visible&lt;3-&gt;{
\node[rectangle,draw=ublue,red,inner sep=0.1em,fill=white] [fit = (eq2) (eq2part2) (eq2part3) (eq2part4)] (eq2full) {};
}
\end{pgfonlayer}

\visible&lt;2-&gt;{
\draw [-&gt;,thick] ([xshift=-3em]exampleeq.south) .. controls +(south:1.5em) and +(north:1.5em) .. (eq1full.north);
}
\visible&lt;3-&gt;{
\draw [-&gt;,thick] ([xshift=3em]exampleeq.south) .. controls +(south:1.5em) and +(north:1.5em) .. (eq2full.north);
}

\visible&lt;4-&gt;{
\node [anchor=west] at ([xshift=0.7em]eq1full.east) {\LARGE{\textbf{=}}};
}

\visible&lt;5-&gt;{
{\large
\node [anchor=west] (feq) at ([xshift=3em,yshift=-3em]eq1full.south west) {$\sum\limits_{a_1=0}^{l} ... \sum\limits_{a_m=0}^{l} \prod\limits_{j=1}^{m}$};
\node [anchor=west,inner sep=2pt,fill=blue!20] (feqpart2) at ([xshift=-0.3em]feq.east) {$f(s_j|t_{a_j})$};
\node [anchor=west,inner sep=1pt] (feqpart3) at (feqpart2.east) {=};
\node [anchor=west] (feqpart4) at (feqpart3.east) {$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l}$};
\node [anchor=west,inner sep=2pt,fill=blue!20] (feqpart5) at ([xshift=-0.3em]feqpart4.east) {$f(s_j|t_i)$};
}

\draw [-&gt;,thick] (eq1part4.south) .. controls +(south:2.5em) and +(north:2.5em) .. (feqpart2.north);
\draw [-&gt;,thick] (eq2part5.south) .. controls +(south:1.5em) and +(north:1.5em) .. (feqpart5.north);

\node [anchor=west,inner sep=2pt,fill=blue!20] (eq1part4) at ([xshift=-0.3em]eq1part3.east) {\footnotesize{$\alpha(x,y_x)$}};
\node [anchor=west,inner sep=2pt,fill=blue!20] (eq2part5) at ([xshift=-0.3em]eq2part4.east){\footnotesize{$\alpha(x,y)$}};

}

\end{tikzpicture}
\end{center}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
\subsection{ѵ��}

%%%------------------------------------------------------------------------------------------------------------
%%% outline
\begin{frame}{Outline}

\begin{center}
\begin{tikzpicture}

{\Large
\node [anchor=west,draw,gray,very thick,rounded corners=4pt,text width=15em,align=left,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (p1) at (0,0) {\gray{\textbf{1. һ���򵥵ķ���ʵ��}}};

\node [anchor=north west,draw,ublue,very thick,rounded corners=4pt,text width=15em,align=left,fill=white,drop shadow={shadow xshift=0.1em,shadow yshift=-0.1em}] (p21) at ([yshift=-1em]p1.south west) {\black{\textbf{2. IBMģ��}}\\\gray{\textbf{\hspace{1.5em}$\bullet$  ��ģ}}\\\gray{\textbf{\hspace{1.5em}$\bullet$  ����}}\\\black{\textbf{\hspace{1.5em}$\bullet$  ģ��ѵ��}}};
}

\end{tikzpicture}
\end{center}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% objectives of training
\begin{frame}{ѵ�� - Ŀ�꺯��}

\begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true]{\textbf{ѵ�� = �Ż�}}
��ѵ�����ϵ�������ʹ��\alert{Ŀ�꺯��}��ֵ�ﵽ�����С������ʱ�Ĳ���������ģ���ڸ�Ŀ�꺯���µ�\alert{���Ž�}

\vspace{-0.5em}

\begin{center}
\begin{tikzpicture}

\begin{scope}[scale=0.8]

\draw [-,very thick] (0,0) sin (1,1) cos (2,0) sin (3,-1) cos (4,0) sin (7,-1);
\draw [-latex,thick] (-0.5,-1.2) -- (8,-1.2);
\draw [-latex,thick] (-0.5,-1.2) -- (-0.5,1.3);
\draw [-,dashed] (1,1) -- (1,-1.2);

\node [anchor=north] at (1,-1.2) {\scriptsize{����$x$�����Ž�}};
\node [anchor=center] at (1,-1.2) {$\bullet$};
\node [anchor=center] at (1,1) {$\bullet$};
\node [anchor=west] at (1.3,1) {\scriptsize{Ŀ�꺯��$f(x)$�����ֵ}};
\draw [&lt;-] (6.5,-0.8) -- (7.0,-0.3);
\node [anchor=south west,inner sep=1pt] at (7.0,-0.3) {\scriptsize{$f(x)$}};

\end{scope}

\end{tikzpicture}
\end{center}

\end{beamerboxesrounded}

\begin{itemize}

\item&lt;2-&gt; \textbf{IBMģ�͵�ѵ�������ڸ����ľ��$(s,t)$����󻯷������$\textrm{P}(s|t)$}�������÷���$\textrm{P}_{\theta}(s|t)$��ʾ�����ɲ���$\theta$����

\begin{center}
\begin{tikzpicture}

\node [anchor=west] (eq1) at (0,0) {$\hat{\theta}$};
\node [anchor=west] (eq2) at ([yshift=-0.2em]eq1.east) {=};
\node [anchor=west,inner sep=2pt] (eq3) at ([yshift=-0.0em]eq2.east) {$\argmax$};
\node [anchor=north,inner sep=1pt] (eq3part2) at ([yshift=-0.2em]eq3.south) {\scriptsize{$\theta$}};
\node [anchor=west,inner sep=2pt] (eq4) at ([xshift=0.1em]eq3.east) {$\textrm{P}_{\theta}(s|t)$};

\visible&lt;3-&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=1.35em] (eq3) at ([yshift=-0.0em]eq2.east) {$\argmax$};
\node [anchor=west,inner sep=2pt,fill=green!20] (eq4) at ([xshift=0.1em]eq3.east) {$\textrm{P}_{\theta}(s|t)$};

\node [anchor=north,draw,inner sep=3pt,fill=red!20] (eq3label) at ([yshift=-1.5em]eq3.south west) {\footnotesize{\textbf{�����Ų���}}};
\node [anchor=north,draw,inner sep=3pt,fill=green!20] (eq4label) at ([yshift=-1.5em]eq4.south east) {\footnotesize{\textbf{Ŀ�꺯��}}};

\draw [-&gt;,thick] ([xshift=-1em]eq3.south) .. controls +(south:1em) and +(north:1em) .. (eq3label.north);
\draw [-&gt;,thick] (eq4.south) .. controls +(south:1em) and +(north:1em) .. (eq4label.north);
}

\end{tikzpicture}
\end{center}


\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% maximizing the likelihood
\begin{frame}{������Ȼ����}

\begin{itemize}

\item $\textrm{P}(s|t)$���Ա�������$(s,t)$�ϵ�\alert{��Ȼ}����($L(s,t;\theta)$)����ν\alert{������Ȼ����}������Ҫ�ҵ�ʹ$L(s,t;\theta)$�ﵽ����$\theta$��

    \vspace{-0.5em}

    \begin{displaymath}
    \{\hat{\theta}\} \subseteq \{\argmax_{\theta \in \Theta} L(s,t;\theta)\}
    \end{displaymath}

    \vspace{-0.3em}

    $L(s,t;\theta)$��ʾ$L(\cdot)$����ģ�Ͳ���$\theta$��ע��ֺţ���$\{\hat{\theta}\}$��ʾ�����ж�������$\Theta$��ʾ�����ռ�

\vspace{0.5em}

\item&lt;2-&gt; �Ȳ��ÿ�������Ĺ�ʽ�����ǻ��ǻع鵽ԭʼ���⣺����ҵ�һ��$\theta$ʹ$\textrm{P}_{\theta}(s|t)$�ﵽ���\\
    \begin{itemize}
    \item \textbf{�������ֵ����}�����磬���ǿ��Զ�$\textrm{P}_{\theta}(s|t)$�󵼣����Ϊ�㣬�õ���ֵ��
    \end{itemize}

\end{itemize}

\vspace{-0.5em}

\visible&lt;2-&gt;{
\begin{center}
\begin{tikzpicture}

\begin{scope}[scale=0.18]
\draw[-,thick,ublue,domain=-4.2:3.5,samples=100] plot (\x,{ - 1/14 * (\x + 4) * (\x + 1) * (\x - 1) * (\x - 3)});
\draw[-&gt;] (-6,0) -- (5,0);
\draw[-&gt;] (-5,-5) -- (-5,5);
\node[anchor=north,inner sep=1pt] at (4.5,-0.3) {\tiny{$\theta$}};
\node[anchor=west,inner sep=1pt] (label1) at (2.0,4.5) {\tiny{$g(\theta)$}};
\draw[-&gt;] ([xshift=-3em]label1.south) -- (2.3,2);
\end{scope}

\draw[-&gt;,double,thick] (0.6in,0) -- (0.9in,0) node [pos=0.5,below,yshift=-3pt] {\tiny{\textbf{��}}};

\begin{scope}[scale=0.18,xshift=8.5in]
\draw[-,thick,ublue,domain=-3.5:3.0,samples=100] plot (\x,{ - 1/14 * (4*\x*\x*\x + 3*\x*\x - 26*\x - 1)});
\draw[-&gt;] (-6,0) -- (5,0);
\draw[-&gt;] (-5,-5) -- (-5,5);
\node[anchor=north,inner sep=1pt] at (4.5,-0.3) {\tiny{$\theta$}};
\node[anchor=west,inner sep=1pt] (label1) at (2.0,4.5) {\tiny{$\frac{\partial g(\theta)}{\partial \theta}$}};
\draw[-&gt;] ([xshift=-3em]label1.south) -- (2.3,2);
\node [red] at (0,0) {\tiny{$\bullet$}};
\node [red] at (-3.0,0) {\tiny{$\bullet$}};
\node [red] at (2.2,0) {\tiny{$\bullet$}};
\end{scope}

\draw[-&gt;,double,thick] (2.1in,0) -- (2.4in,0) node [pos=0.5,below,yshift=-3pt] (ltmp) {\tiny{\textbf{��$\frac{\partial g(\theta)}{\partial \theta}=0$}}};
\node [anchor=north west] at ([yshift=5pt]ltmp.south west) {\tiny{\textbf{��ֵ��}}};

\begin{scope}[scale=0.18,xshift=17in]
\draw[-,thick,ublue,domain=-4.2:3.5,samples=100] plot (\x,{ - 1/14 * (\x + 4) * (\x + 1) * (\x - 1) * (\x - 3)});
\draw[-,dashed,ublue,domain=-3.8:3.0,samples=100] plot (\x,{ - 1/14 * (4*\x*\x*\x + 3*\x*\x - 26*\x - 1)});
\draw[-&gt;] (-6,0) -- (5,0);
\draw[-&gt;] (-5,-5) -- (-5,5);
\node[anchor=north,inner sep=1pt] at (4.5,-0.3) {\tiny{$\theta$}};

%\node [red] at (0,0) {\tiny{$\bullet$}};
\node [red] at (-3.0,0) {\tiny{$\bullet$}};
%\node [red] at (2.2,0) {\tiny{$\bullet$}};
\draw[-,red] (-3,0)--(-3,3.3);
\node [red] at (-3.0,3.3) {\tiny{$\bullet$}};
\node[anchor=south,inner sep=1pt] (label1) at (-3.0,6) {\tiny{\textbf{���ֵ}}};
\draw[-&gt;] ([xshift=1em]label1.south) -- (-3.0,4);

\node[anchor=south,inner sep=1pt] (label2) at (-3.0,-5) {\tiny{\textbf{����$\theta$}}};
\draw[-&gt;] ([xshift=-1em]label2.north) -- (-3.1,-1.0);
\end{scope}

\end{tikzpicture}
\end{center}
}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% maximizing P(s|t)
\begin{frame}{���$\textrm{P}(s|t)$}

\begin{itemize}

\item ��IBMģ��1Ϊ����������Ӧ��(���ڼ�����Ȼ���ƣ���\alert{ģ��ѵ������}���Ա�����Ϊ��

\vspace{-0.8em}

\begin{center}
\begin{tikzpicture}

{\large
\node [anchor=west,minimum height=3em,inner sep=2pt] (eq11) at (0,0) {max};
\node [anchor=west,inner sep=2pt] (eq12) at ([xshift=2pt]eq11.east) {$\Big( \frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l}$};
\node [anchor=west,inner sep=2pt] (eq13) at (eq12.east) {$f(s_j|t_i)$};
\node [anchor=west,inner sep=2pt] (eq14) at ([yshift=0.2em]eq13.east) {$\Big)$};

\node [anchor=north west,minimum height=1.8em,inner sep=2pt] (eq21) at ([yshift=-0.5em]eq11.south west) {s.t.};
\node [anchor=west] (eq22) at ([xshift=0.5em,yshift=-0.1em]eq21.east) {\normalsize{for each word $t_y$}:};
\node [anchor=west] (eq23) at ([xshift=0.1em,yshift=-0.0em]eq22.east) {$\sum_{s_x} f(s_x|t_y) = 1$};

\visible&lt;2-&gt;{
\node [anchor=west,minimum height=3em,inner sep=2pt,fill=green!20] (eq11) at (0,0) {max};
\node [anchor=west,draw,thin,ublue,inner sep=2pt,fill=blue!20] (eq13) at (eq12.east) {\black{$f(s_j|t_i)$}};

\node [anchor=north west,minimum height=1.8em,inner sep=2pt,fill=purple!20] (eq21) at ([yshift=-0.5em]eq11.south west) {s.t.};
\node [anchor=west,fill=orange!20] (eq23) at ([xshift=0.1em,yshift=-0.0em]eq22.east) {$\sum_{s_x} f(s_x|t_y) = 1$};
}

\begin{pgfonlayer}{background}
\visible&lt;2-&gt;{
\node[rectangle,red,inner sep=0pt,fill=red!20] [fit = (eq12) (eq13) (eq14)] (eq1full) (eq1obj) {};
}
\end{pgfonlayer}
}

\visible&lt;2-&gt;{
\node [anchor=west,draw,fill=green!20] (label1) at ([xshift=-2em,yshift=3em]eq11.north west) {��󻯺���};
\node [anchor=west,draw,fill=red!20] (label2) at ([xshift=4em]label1.east) {Ŀ�꺯��};
\node [anchor=west,draw,fill=blue!20] (label3) at ([xshift=4em]label2.east) {����};

\node [anchor=west,draw,fill=purple!20] (label4) at ([xshift=-2em,yshift=-2.5em]eq21.south west) {subject to=����...Լ��};
\node [anchor=west,draw,fill=orange!20] (label5) at ([xshift=3em]label4.east) {������ʹ�һ��Լ������};

\draw [&lt;-,thick] ([xshift=-2em]label1.south) .. controls +(south:2em) and +(north:2em) .. (eq11.north);
\draw [&lt;-,thick] ([xshift=-0em]label2.south) .. controls +(south:1.7em) and +(north:1.7em) .. (eq1obj.north);
\draw [&lt;-,thick] ([xshift=-0em]label3.south) .. controls +(south:2.5em) and +(north:2.5em) .. (eq13.north);
\draw [&lt;-,thick] ([xshift=2em]label4.north) .. controls +(north:1.5em) and +(south:1.5em) .. (eq21.south);
\draw [&lt;-,thick] ([xshift=0em]label5.north) .. controls +(north:1.0em) and +(south:1.0em) .. (eq23.south);
}

\end{tikzpicture}
\end{center}

    \begin{itemize}
    \item&lt;2-&gt; \alert{ע��}��\{$f(s_x|t_y)$\}��Ӧ�ܶ������ÿ��Դ���Ե��ʺ�ÿ��Ŀ���ﵥ�ʵ���϶���Ӧһ��$f(s_x|t_y)$
    \end{itemize}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% optimizaton -  Lagrange multipliers
\begin{frame}{�������ճ����� - Lagrange multiplier}

\begin{itemize}

\item \textbf{����Լ�����Ż�����}: ���ý�\\
      \textbf{Ŀ��:} $\max(\textrm{P}_{\theta}(s|t))$ + \textbf{Լ��:} $\forall t_y: \sum_{s_x} \textrm{P}(s_x|t_y)=1$

\vspace{0.3em}
\item&lt;2-&gt; \textbf{�������}: ����Լ���Ż� $\Rightarrow$ ����Լ���Ż�\\
\vspace{0.3em}
\begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true,width=\linewidth]{\textbf{�������ճ����� - �ٶȰٿ�}}

{\small
      \textit{\alert{�������ճ�����}��һ��Ѱ�ұ�����һ���������������ƵĶ�Ԫ�����ļ�ֵ�ķ��������ַ�����һ����$n$��������$k$��Լ�����������Ż�����ת��Ϊһ����$n+ k$�������ķ�����ļ�ֵ���⣬����������κ�Լ�������ַ���������һ���µı���δ֪�������������ճ�����}
}
\end{beamerboxesrounded}

    \begin{itemize}
    \item ����ÿ��$t_y$����Ӧ��Լ��������һ���������ճ���$\lambda_{t_y}$
    \end{itemize}

    \vspace{-1em}

    {\small
    \begin{eqnarray}
    L(f,\lambda) &amp; = &amp; \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 \\
                 &amp;   &amp; \sum_{t_y} \lambda_{t_y} (\sum_{s_x} f(s_x|t_y) -1) \nonumber
    \end{eqnarray}
    }

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% illustration -  Lagrange multipliers
\begin{frame}{�������ճ����� - Lagrange multiplier (2)}

\begin{center}
\begin{tabular}{c | c}
\makebox[0.35\textwidth][c]{\textbf{ԭʼ����}} &amp; \makebox[0.35\textwidth][c]{\textbf{ת���������}} \\ \hline
$\max (\textrm{P}(s|t))$ &amp; $\max (L(f,\lambda))$ \\
s.t. $\forall t_y: \sum_{s_x} f(s_x|t_y) =1 $ &amp; \\
\end{tabular}
\end{center}


\begin{center}
\begin{tikzpicture}

\node [anchor=west,inner sep=2pt,fill=green!20] (eq1) at (0,0) {$L(f,\lambda)$};
\node [anchor=west,inner sep=2pt] (eq2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq3) at (eq2.east) {$\frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l}$};
\node [anchor=west,inner sep=2pt] (eq4) at (eq3.east) {\black{$f(s_j|t_{i})$}};
\node [anchor=west,inner sep=2pt] (eq5) at (eq4.east) {$-$};
\node [anchor=north west,inner sep=2pt] (eq6) at ([yshift=-6pt]eq3.south west) {$\sum_{t_y}$};
\node [anchor=west,inner sep=1pt,minimum height=1.85em] (eq7) at (eq6.east) {$\lambda_{t_y}$};
\node [anchor=west,inner sep=1pt,minimum height=1.5em] (eq8) at ([xshift=3pt]eq7.east) {$(\sum_{s_x}$};
\node [anchor=west,inner sep=1pt,minimum height=1.5em] (eq9) at (eq8.east) {\black{$f(s_x|t_y)$}};
\node [anchor=west,inner sep=1pt,minimum height=1.5em] (eq10) at (eq9.east) {$-1)$};

\visible&lt;2-&gt;{
\node [anchor=west,inner sep=2pt,fill=green!20] (eq1) at (0,0) {$L(f,\lambda)$};
\node [anchor=west,inner sep=2pt] (eq2) at (eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq3) at (eq2.east) {$\frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l}$};
\node [anchor=west,inner sep=2pt,draw,ublue,fill=blue!20] (eq4) at (eq3.east) {\black{$f(s_j|t_{i})$}};
\node [anchor=west,inner sep=2pt] (eq5) at (eq4.east) {$-$};
\node [anchor=north west,inner sep=2pt] (eq6) at ([yshift=-6pt]eq3.south west) {$\sum_{t_y}$};
\node [anchor=west,inner sep=1pt,fill=purple!20,minimum height=1.85em] (eq7) at (eq6.east) {$\lambda_{t_y}$};
\node [anchor=west,inner sep=1pt,minimum height=1.5em] (eq8) at ([xshift=3pt]eq7.east) {$(\sum_{s_x}$};
\node [anchor=west,inner sep=1pt,draw,ublue,fill=blue!20,minimum height=1.5em] (eq9) at (eq8.east) {\black{$f(s_x|t_y)$}};
\node [anchor=west,inner sep=1pt,minimum height=1.5em] (eq10) at (eq9.east) {$-1)$};
}

\begin{pgfonlayer}{background}
\node[rectangle,red,inner sep=2pt,fill=red!20] [fit = (eq3) (eq4)] (oldobj) {};
\visible&lt;2-&gt;{
\node[rectangle,red,inner sep=2pt,fill=orange!20] [fit = (eq8) (eq9) (eq10)] (constraint) {};
}
\end{pgfonlayer}

\node [anchor=west,draw,fill=green!20] (label1) at ([xshift=-2em,yshift=3em]eq1.north west) {��Ŀ�꺯��};
\node [anchor=west,draw,fill=red!20] (label2) at ([xshift=2em]label1.east) {��Ŀ�꺯��};

\visible&lt;2-&gt;{
\node [anchor=west,draw,fill=blue!20] (label3) at ([xshift=2em]label2.east) {�������ʷ������};
\node [anchor=west,draw,fill=purple!20] (label4) at ([xshift=-4em,yshift=-2.5em]eq6.south west) {�������������ճ���};
\node [anchor=west,draw,fill=orange!20] (label5) at ([xshift=3em]label4.east) {����Լ������};
}

\draw [&lt;-,thick] ([xshift=-1em]label1.south) .. controls +(south:2em) and +(north:2em) .. (eq1.north);
\draw [&lt;-,thick] ([xshift=-1em]label2.south) .. controls +(south:1.0em) and +(north:1.0em) .. (oldobj.north);

\visible&lt;2-&gt;{
\draw [&lt;-,thick] ([xshift=0em]label3.south) .. controls +(south:6.5em) and +(north:1.5em) .. (eq9.north);
\draw [&lt;-,thick] ([xshift=0em]label3.south) .. controls +(south:1.5em) and +(north:1.5em) .. (eq4.north);
\draw [&lt;-,thick] ([xshift=-2em]label4.north) .. controls +(north:1.5em) and +(south:1.5em) .. (eq7.south);
\draw [&lt;-,thick] ([xshift=0em]label5.north) .. controls +(north:1.5em) and +(south:1.5em) .. (constraint.south);
}

\end{tikzpicture}
\end{center}

\begin{itemize}
\item&lt;3-&gt; \textbf{ʣ�µ�����}������$\frac{\partial L(f,\lambda)}{\partial f(s_u|t_v)}$�������$\frac{\partial L(f,\lambda)}{\partial f(s_u|t_v)} = 0$
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% d L / d \theta = ?
\begin{frame}{��$L$��}

\begin{itemize}
\item ��$s_u$��$t_v$��ʾ����һ��Դ�ﵥ�ʺ�һ��Ŀ���ﵥ��
\vspace{-0.5em}
\begin{eqnarray}
\frac{\partial L(f,\lambda)}{\partial f(s_u|t_v)} &amp; = &amp; \frac{\partial \big[ \frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \big]}{\partial f(s_u|t_v)} - \nonumber \\
 &amp;   &amp; \frac{\partial \big[ \sum_{t_y} \lambda_{t_y} (\sum_{s_x} f(s_x|t_y) -1) \big]}{\partial f(s_u|t_v)} \nonumber \\
 &amp; = &amp; \frac{\epsilon}{(l+1)^{m}} \cdot \frac{\partial \big[ \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \big]}{\partial f(s_u|t_v)} - \lambda_{t_v} \nonumber
\end{eqnarray}

\vspace{-0.3em}

\item&lt;2-&gt; Ϊ����$\frac{\partial \big[ \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \big]}{\partial f(s_u|t_v)}$���ȿ�һ���򵥵����ӣ� $g(z)=\alpha z^{\beta}$Ϊһ������$z$�Ķ���ʽ��������Ȼ
\vspace{-0.3em}
\begin{displaymath}
\frac{\partial g(z)}{\partial z} = \alpha \beta z^{\beta-1} = \frac{\beta}{z}\alpha z^{\beta} = \frac{\beta}{z} g(z)
\end{displaymath}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% d L / d \theta = ? (continued)
\begin{frame}{��$L$��(2)}
\begin{itemize}
\item ������԰�$\prod_{j=1}^{m} \sum_{i=0}^{l} f(s_j|t_i)$����$g(z)=\alpha z^{\beta}$��ʵ��
    \begin{itemize}
    \item ��$z=\sum_{i=0}^{l}f(s_u|t_i)$��ע��$s_u$Ϊ������Դ�ﵥ��
    \item ��ô$\beta$Ϊ$\sum_{i=0}^{l}f(s_u|t_i)$��$\prod_{j=1}^{m} \sum_{i=0}^{l} f(s_j|t_i)$�г��ֵĴ�������Դ���������$s_u$��ͬ�ĵ��ʵĸ���
    \end{itemize}
    \begin{displaymath}
    \beta=\sum_{j=1}^{m} \delta(s_j,s_u)
    \end{displaymath}
    $\delta(x,y)$����$x=y$ʱΪ1������Ϊ0
    \vspace{0.5em}
\item&lt;2-&gt; ����$\frac{\partial g(z)}{\partial z} = \frac{\beta}{z} g(z)$
    \begin{displaymath}
    \frac{\partial g(z)}{\partial z} = \frac{\partial \big[ \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \big]}{\partial \big[ \sum\limits_{i=0}^{l}f(s_u|t_i) \big]} = \frac{\sum\limits_{j=1}^{m} \delta(s_j,s_u)}{\sum\limits_{i=0}^{l}f(s_u|t_i)} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)
    \end{displaymath}
\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% d L / d \theta = ? (continued)
\begin{frame}{��$L$��(3)}
\begin{itemize}
\item �����$z$����$f$�ĺ�������$\frac{\partial g(z)}{\partial f} = \frac{\partial g(z)}{\partial z} \cdot \frac{\partial z}{\partial f}$
    \begin{itemize}
    \item \textbf{��Ϊ}$z=\sum_{i=0}^{l}f(s_u|t_i)$�������׵õ�
    \begin{displaymath}
    \frac{\partial z}{\partial f} = \frac{\partial \big[ \sum_{i=0}^{l}f(s_u|t_i) \big]}{\partial f(s_u|t_v)} = \sum_{i=0}^{l} \delta(t_i,t_v)
    \end{displaymath}
    ��Ŀ�������ĵ�������$t_v$��ͬ�ĸ���
    \vspace{0.5em}
    \item&lt;2-&gt; \textbf{����}$\frac{\partial g(z)}{\partial z}$��$\frac{\partial z}{\partial f}$����Ľ�������Եõ�
    \begin{center}
    \begin{tikzpicture}
    \node[anchor=west,inner sep=2pt] (eq11) at (0,0) {\large{$\frac{\partial \big[ \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \big]}{\partial f(s_u|t_v)}$}};
    \node[anchor=west,inner sep=2pt] (eq12) at ([yshift=-9pt]eq11.east) {=};
    \node[anchor=west,inner sep=2pt,fill=red!20] (eq13) at ([yshift=2pt]eq12.east) {\large{$\frac{\partial \big[ \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \big]}{\partial \big[ \sum\limits_{i=0}^{l}f(s_u|t_i) \big]}$}};
    \node[anchor=west,inner sep=2pt] (eq14) at ([yshift=-1pt]eq13.east) {$\cdot$};
    \node[anchor=west,inner sep=2pt,fill=blue!20,minimum height=4em] (eq15) at ([yshift=8pt]eq14.east) {\large{$\frac{\partial \big[ \sum\limits_{i=0}^{l}f(s_u|t_i) \big]}{\partial f(s_u|t_v)}$}};
    \node[anchor=south,inner sep=1pt] (label1) at ([yshift=1pt]eq13.north) {\footnotesize{$\partial g(z)/\partial z$}};
    \node[anchor=south,inner sep=1pt] (label2) at ([yshift=1pt]eq15.north) {\footnotesize{$\partial z/\partial f$}};
    \node[anchor=north west,inner sep=2pt] (eq21) at (eq13.south west) {$\frac{\sum\limits_{j=1}^{m} \delta(s_j,s_u)}{\sum\limits_{i=0}^{l}f(s_u|t_i)} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \cdot \sum\limits_{i=0}^{l} \delta(t_i,t_v)$};
    \node[anchor=east,inner sep=2pt] (eq20) at (eq21.west) {=};
    \end{tikzpicture}
    \end{center}
    \end{itemize}
\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% d L / d \theta = ? (continued)
\begin{frame}{��$L$��(4)}
\begin{itemize}
\item ��$\frac{\partial \big[ \prod_{j=1}^{m} \sum_{i=0}^{l} f(s_j|t_i) \big]}{\partial f(s_u|t_v)}$��һ������$\frac{\partial L(f,\lambda)}{\partial f(s_u|t_v)}$
\end{itemize}
{\small
\begin{eqnarray}
&amp;   &amp; \frac{\partial L(f,\lambda)}{\partial f(s_u|t_v)} \nonumber \\
&amp; = &amp; \frac{\epsilon}{(l+1)^{m}} \cdot \alert{\frac{\partial \big[ \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_{a_j}) \big]}{\partial f(s_u|t_v)}} - \lambda_{t_v} \nonumber \\
&amp; = &amp; \frac{\epsilon}{(l+1)^{m}} \cdot \alert{\frac{\sum_{j=1}^{m} \delta(s_j,s_u) \cdot \sum_{i=0}^{l} \delta(t_i,t_v)}{\sum_{i=0}^{l}f(s_u|t_i)} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)} - \lambda_{t_v} \nonumber
\end{eqnarray}
}

\begin{itemize}
\item&lt;2-&gt; ��$\frac{\partial L(f,\lambda)}{\partial f(s_u|t_v)}=0$����
\end{itemize}

\vspace{-1em}

\visible&lt;2-&gt;{
{\small
\begin{displaymath}
f(s_u|t_v) = \frac{\lambda_{t_v}^{-1} \epsilon}{(l+1)^{m}} \cdot \frac{\sum\limits_{j=1}^{m} \delta(s_j,s_u) \cdot \sum\limits_{i=0}^{l} \delta(t_i,t_v)}{\sum\limits_{i=0}^{l}f(s_u|t_i)} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i) \cdot f(s_u|t_v)
\end{displaymath}
}
}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% learning P(s_u |t_v) - EM
\begin{frame}{��$f(s_u |t_v)$�����Ž� - �ⲻ��һ������ʽ������}
\begin{itemize}
\item \alert{ע��}���ⲻ��һ������$f(s_u |t_v)$�Ľ���ʽ����Ϊ��ʽ�Ҷ��Ժ���$f(s_u |t_v)$
\end{itemize}

\vspace{-1em}

\begin{center}
\begin{tikzpicture}

{\small
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3em] (eq1) at (0,0) {$f(s_u|t_v)$};
\node [anchor=west,inner sep=2pt] (eq2) at ([xshift=-2pt]eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq3) at ([xshift=-2pt]eq2.east) {$\lambda_{t_v}^{-1}$};
\node [anchor=west,inner sep=2pt] (eq4) at ([xshift=-2pt]eq3.east) {$\frac{\epsilon}{(l+1)^{m}}$};
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3em] (eq5) at ([xshift=-2pt]eq4.east) {\footnotesize{$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$}};
\node [anchor=west,inner sep=2pt] (eq6) at ([xshift=-2pt]eq5.east) {\footnotesize{$\sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v)$}};
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3em] (eq7) at ([xshift=-2pt,yshift=-0pt]eq6.east) {$\frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$};
}

\end{tikzpicture}
\end{center}

\vspace{-0.5em}

\begin{itemize}
\item&lt;2-&gt; \textbf{��ô��?} - �����Կ���������ʽ�����ؼ���$f(s_u |t_v)$��ʹ����������������ֵ
    \begin{itemize}
    \item ����һ���ǳ�����ķ����������������(Expectation Maximization)�����EM����(���㷨)
    \item \textbf{˼��}���õ�ǰ�Ĳ�������һ����Ȼ������������֮������������ֵͬʱ�õ��µ�һ�������ֵ
    \item \textbf{����IBMģ�ͣ��ܼ�}������ʹ����ʽ���ɣ�
    \end{itemize}
\end{itemize}

\vspace{-1.3em}

\visible&lt;2-&gt;{
\begin{center}
\begin{tikzpicture}

\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3em] (eq1) at (0,0) {$f(s_u|t_v)$};
\node [anchor=west,inner sep=2pt] (eq2) at ([xshift=-2pt]eq1.east) {$=$};
\node [anchor=west,inner sep=2pt] (eq3) at ([xshift=-2pt]eq2.east) {$\lambda_{t_v}^{-1}$};
\node [anchor=west,inner sep=2pt] (eq4) at ([xshift=-2pt]eq3.east) {$\frac{\epsilon}{(l+1)^{m}}$};
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3em] (eq5) at ([xshift=-2pt]eq4.east) {\footnotesize{$\prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$}};
\node [anchor=west,inner sep=2pt] (eq6) at ([xshift=-2pt]eq5.east) {\footnotesize{$\sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v)$}};
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3em] (eq7) at ([xshift=-2pt,yshift=-0pt]eq6.east) {$\frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$};

\node [anchor=south west,inner sep=2pt] (label1) at ([yshift=1em]eq1.north west) {\footnotesize{\textbf{�µĲ���ֵ}}};
\node [anchor=south east,inner sep=2pt] (label2) at ([yshift=1em,xshift=-5em]eq7.north east) {\footnotesize{\textbf{�ɵĲ���ֵ}}};


\draw [&lt;-,thick] (label1.south) .. controls +(south:1em) and +(north:1em) .. ([xshift=-1em]eq1.north);
\draw [&lt;-,thick] (label2.south) .. controls +(300:1em) and +(north:1em) .. ([xshift=1em]eq7.north);
\draw [&lt;-,thick] ([xshift=-0.5em]label2.south) .. controls +(240:1em) and +(north:1em) .. ([xshift=1em]eq5.north);

\visible&lt;3-&gt;{
\node [anchor=west,inner sep=2pt,fill=blue!20,minimum height=3em] (eq3) at ([xshift=-2pt]eq2.east) {$\lambda_{t_v}^{-1}$};
\node [anchor=west,inner sep=2pt,fill=blue!20] (label3) at ([xshift=2em,yshift=-0.1em]label1.east) {\footnotesize{\textbf{���⣺$\lambda_{t_v}=?$}}};
\draw [&lt;-,thick] (label3.south) .. controls +(south:1em) and +(north:1em) .. (eq3.north);
}

\end{tikzpicture}
\end{center}
}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% learning P(s_u |t_v) - a simiplied form
\begin{frame}{��$f(s_u |t_v)$�ļ�����л���}

\begin{itemize}
\item Ϊ�˼���$f(s_u |t_v)$��������֯һ������Ĺ�ʽ
\end{itemize}

\vspace{-2.0em}

\begin{center}
\begin{tikzpicture}

\node [anchor=west,inner sep=2pt,minimum height=2em] (eq1) at (0,0) {$f(s_u|t_v)$};
\node [anchor=west,inner sep=2pt] (eq2) at ([xshift=-2pt]eq1.east) {$=$};
\node [anchor=west,inner sep=2pt,minimum height=2em] (eq3) at ([xshift=-2pt]eq2.east) {$\lambda_{t_v}^{-1}$};
\node [anchor=west,inner sep=2pt,minimum height=3.0em] (eq4) at ([xshift=-3pt]eq3.east) {\footnotesize{$\frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$}};
\node [anchor=west,inner sep=2pt,minimum height=3.0em] (eq5) at ([xshift=1pt]eq4.east) {\footnotesize{$\sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v)$}};
\node [anchor=west,inner sep=2pt,minimum height=3.0em] (eq6) at ([xshift=1pt]eq5.east) {$\frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$};


\visible&lt;2-&gt;{
\node [anchor=west,inner sep=2pt,fill=red!20,minimum height=3.0em] (eq4) at ([xshift=-3pt]eq3.east) {\footnotesize{$\frac{\epsilon}{(l+1)^{m}} \prod\limits_{j=1}^{m} \sum\limits_{i=0}^{l} f(s_j|t_i)$}};
}
\visible&lt;3-&gt;{
\node [anchor=west,inner sep=2pt,fill=blue!20,minimum height=3.0em] (eq5) at ([xshift=1pt]eq4.east) {\footnotesize{$\sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v)$}};
}
\visible&lt;4-&gt;{
\node [anchor=west,inner sep=2pt,fill=green!20,minimum height=3.0em] (eq6) at ([xshift=1pt]eq5.east) {$\frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$};
}

\visible&lt;2-&gt;{
\node [anchor=south west,inner sep=2pt] (label1) at (eq4.north west) {\textbf{\scriptsize{�������$\textrm{P}(s|t)$}}};
}
\visible&lt;3-&gt;{
\node [anchor=south west,inner sep=2pt] (label2) at (eq5.north west) {\textbf{\scriptsize{��Ե��ܴ���}}};
\node [anchor=south west,inner sep=2pt] (label2part2) at ([yshift=-3pt]label2.north west) {\textbf{\scriptsize{$(s_u,t_v)$�ھ��$(s,t)$��}}};
}
\visible&lt;4-&gt;{
\node [anchor=south west,inner sep=2pt] (label3) at (eq6.north west) {\textbf{\scriptsize{�е�$t_i$�����ֵ}}};
\node [anchor=south west,inner sep=2pt] (label4) at ([yshift=-3pt]label3.north west) {\textbf{\scriptsize{$f(s_u|t_v)$������}}};
}

\visible&lt;2-&gt;{
\node [anchor=east,rotate=90] (neweq1) at ([yshift=-0em]eq4.south) {=};
\node [anchor=north,inner sep=1pt] (neweq1full) at (neweq1.west) {\large{$\textrm{P}(s|t)$}};
}

\visible&lt;5-&gt;{
\draw[decorate,thick,decoration={brace,amplitude=5pt,mirror}] ([yshift=-0.2em]eq5.south west) -- ([yshift=-0.2em]eq6.south east) node [pos=0.4,below,xshift=-0.0em,yshift=-0.3em] (expcount1) {\footnotesize{\textbf{'$t_v$����Ϊ$s_u$'����¼�}}};
\node [anchor=north west] (expcount2) at ([yshift=0.5em]expcount1.south west) {\footnotesize{\textbf{���ִ����������Ĺ���}}};
\node [anchor=north west] (expcount3) at ([yshift=0.5em]expcount2.south west) {\footnotesize{\textbf{��֮Ϊ����Ƶ��expected count}}};
}

\end{tikzpicture}
\end{center}

\vspace{-1.2em}

\begin{itemize}
\item&lt;6-&gt; \textbf{ɶ���¼�������Ƶ�Σ�����} - �¼�����ֲ��³��ֵĴ���������$c_{\mathbb{E}}(X) = \sum_i c(x_i) \cdot \textrm{P}(x_i)$
\end{itemize}

\vspace{-0.8em}

\visible&lt;6-&gt;{
\begin{center}
\begin{minipage}[t]{0.30\linewidth}
\begin{tabular}{c | c}
$x_i$ &amp; $c(x_i)$ \\ \hline
$x_1$ &amp; 2 \\
$x_2$ &amp; 1 \\
$x_3$ &amp; 5 \\ \hline
\multicolumn{2}{c}{\visible&lt;7-&gt;{$c(X)=8$}}
\end{tabular}
\end{minipage}
\begin{minipage}[t]{0.50\linewidth}
\begin{tabular}{c | c | c | c}
$x_i$ &amp; $c(x_i)$ &amp; $\textrm{P}(x_i)$ &amp; $c(x_i) \cdot \textrm{P}(x_i)$\\ \hline
$x_1$ &amp; 2 &amp; 0.1 &amp; 0.2 \\
$x_2$ &amp; 1 &amp; 0.3 &amp; 0.3 \\
$x_3$ &amp; 5 &amp; 0.2 &amp; 1.0 \\ \hline
\multicolumn{4}{c}{\visible&lt;8-&gt;{$c_{\mathbb{E}}(X)=0.2+0.3+1.0=1.5$}}
\end{tabular}
\end{minipage}
\end{center}
}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% introducing the expected counts
\begin{frame}{����Ƶ��}

\begin{center}
\begin{tikzpicture}

\node [anchor=west,inner sep=2pt,minimum height=3.0em] (eq5) at (0,0) {$\sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v)$};
\node [anchor=west,inner sep=2pt,minimum height=3.0em,minimum width=9.0em] (eq6) at ([xshift=1pt]eq5.east) {\Large{$\frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}};

\visible&lt;2-&gt;{
\node [anchor=west,inner sep=2pt,minimum height=3.0em,fill=blue!20] (eq5) at (0,0) {$\sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v)$};
}
\visible&lt;3-&gt;{
\node [anchor=west,inner sep=2pt,minimum height=3.0em,minimum width=9.0em,fill=green!20] (eq6) at ([xshift=1pt]eq5.east) {\Large{$\frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}};
}

\visible&lt;2-&gt;{
\node [anchor=north] (label1) at (eq5.south) {\footnotesize{'$t_v$����Ϊ$s_u$'������}};
\node [anchor=north west] (label1part2) at ([yshift=6pt]label1.south west) {\footnotesize{�����г��ֵĴ���}};
}
\visible&lt;3-&gt;{
\node [anchor=north] (label2) at ([xshift=-1em]eq6.south) {\footnotesize{'$t_v$����Ϊ$s_u$'������}};
\node [anchor=north west] (label2part2) at ([yshift=6pt]label2.south west) {\footnotesize{�����г��ֵ���Ը���}};
}

\visible&lt;2-&gt;{
\node [anchor=north] (c) at (label1part2.south) {$c$('$t_v$����Ϊ$s_u$')};
}
\visible&lt;3-&gt;{
\node [anchor=north] (p) at (label2part2.south) {$\textrm{P}$('$t_v$����Ϊ$s_u$')};
}

\visible&lt;4-&gt;{
\node [anchor=west] (times) at ([xshift=0.3em]c.east) {$\times$};
\node [anchor=north east] (eqeq) at (c.south west) {=};
\node [anchor=west] (eqfinal) at (eqeq.east) {$c_{\mathbb{E}}$('$t_v$����Ϊ$s_u$')};
}

\end{tikzpicture}
\end{center}

\begin{itemize}
\item&lt;5-&gt; \textbf{����}����$\textrm{P}(s|t)$�У�$t_v$����(����)��$s_u$������Ƶ��Ϊ
    \vspace{-0.5em}
    \begin{displaymath}
    c_{\mathbb{E}}(s_u|t_v;s,t) \equiv \sum\limits_{j=1}^{m} \delta(s_j,s_u) \sum\limits_{i=0}^{l} \delta(t_i,t_v) \cdot \frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}
    \end{displaymath}
    \vspace{-0.8em}
\item&lt;6-&gt; \textbf{��д$f(s_u|t_v)$}!!!
    \begin{center}
    \begin{tikzpicture}
    \node [anchor=west,draw,red,thick,inner sep=5pt] (p) at (0,0) {\black{$f(s_u|t_v) = \lambda_{t_v}^{-1} \cdot \textrm{P}(s|t) \cdot c_{\mathbb{E}}(s_u|t_v;s,t)$}};
    \end{tikzpicture}
    \end{center}
\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% calculating f(s_u|t_v) by expected counts
\begin{frame}{ͨ������Ƶ�μ���$f(s_u|t_v)$}

\begin{itemize}
\item \textbf{һ��Сtrick}: ��$\lambda_{t_v}^{'}=\frac{\lambda_{t_v}}{\textrm{P}(s|t)}$

\vspace{-1.0em}

\begin{eqnarray}
f(s_u|t_v) &amp; = &amp; \lambda_{t_v}^{-1} \cdot \textrm{P}(s|t) \cdot c_{\mathbb{E}}(s_u|t_v;s,t) \nonumber \\
                    &amp; = &amp; (\lambda_{t_v}^{'})^{-1} \cdot c_{\mathbb{E}}(s_u|t_v;s,t) \nonumber
\end{eqnarray}

\item&lt;2-&gt; \textbf{$\lambda_{t_v}^{'}$������ʲô��} - ����һ��IBMģ�Ͷ�$f(\cdot|\cdot)$��Լ��
\begin{displaymath}
\forall t_y: \sum_{s_x} f(s_x|t_y) = 1
\end{displaymath}
\vspace{-0.3em}
Ϊ������$f(\cdot|\cdot)$�ĸ��ʹ�һ��Լ������Ȼ

\vspace{-0.3em}

\begin{displaymath}
\lambda_{t_v}^{'}=\sum_{s_u} c_{\mathbb{E}}(s_u|t_v;s,t)
\end{displaymath}

\vspace{-0.6em}

\item&lt;3-&gt; \textbf{$f(s_u|t_v)$�ļ���ʽ}
    \begin{center}
    \begin{tikzpicture}
    \node [anchor=west] (eq1) at (0,0) {$f(s_u|t_v) =$};
    \draw [-] (eq1.east) -- ([xshift=8em]eq1.east);
    \node [anchor=south west] (eq2) at ([xshift=1em]eq1.east) {$c_{\mathbb{E}}(s_u|t_v;s,t)$};
    \node [anchor=north west] (eq3) at (eq1.east) {$\sum_{s_u} c_{\mathbb{E}}(s_u|t_v;s,t)$};

    \begin{pgfonlayer}{background}
    \node[rectangle,draw,red,thick,inner sep=0] [fit = (eq1) (eq2) (eq3)] {};
    \end{pgfonlayer}
    \end{tikzpicture}
    \end{center}

\end{itemize}

\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% scale it up to the full corpus
\begin{frame}{���������ݼ��ϼ���}
\begin{itemize}
\item \textbf{����ʵ�����}������ӵ��һϵ�л���ľ�ԣ�����\alert{ƽ������}������Ϊ$\{(s^{[1]},t^{[1]}),(s^{[2]},t^{[2]}),...,(s^{[N]},t^{[N]})\}$��������$N$��ѵ���þ�ԣ�����$f(s_u|t_v)$������Ƶ��Ϊ
    \begin{displaymath}
    c_{\mathbb{E}}(s_u|t_v) = \sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})
    \end{displaymath}
\item&lt;2-&gt; \textbf{����}
    \begin{center}
    \begin{tikzpicture}
    \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) {$\sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})$};
    \node [anchor=north west] (eq4) at (eq2.east) {$\sum_{s_u} \sum_{i=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[i]},t^{[i]})$};

    \visible&lt;4-&gt;{
    \node [anchor=south] (label1) at ([yshift=-6em,xshift=3em]eq1.north west) {���������ʽ����};
    \node [anchor=north west] (label1part2) at ([yshift=0.3em]label1.south west) {�µ�$f(s_u|t_v)$ֵ};
    }
    \visible&lt;3-&gt;{
    \node [anchor=west] (label2) at ([xshift=5em]label1.east) {�õ�ǰ��$f(s_u|t_v)$};
    \node [anchor=north west] (label2part2) at ([yshift=0.3em]label2.south west) {��������Ƶ��$c_{\mathbb{E}}(\cdot)$};
    }

    \visible&lt;4-&gt;{
    \node [anchor=west,fill=red!20,inner sep=2pt] (eq1) at (0,0) {$f(s_u|t_v)$};
    }

    \begin{pgfonlayer}{background}
    \visible&lt;3-&gt;{
    \node[rectangle,fill=blue!20,inner sep=0] [fit = (eq3) (eq4)] (c) {};
    }
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    \node[rectangle,draw,red,inner sep=0] [fit = (label1) (label1part2)] (flabel) {};
    }
    \visible&lt;3-&gt;{
    \node[rectangle,draw,ublue,inner sep=0] [fit = (label2) (label2part2)] (clabel) {};
    }
    \end{pgfonlayer}

    \visible&lt;4-&gt;{
    \draw [-&gt;,thick] (eq1.south) ..controls +(south:1.5em) and +(north:1.5em).. (flabel.north);
    }
    \visible&lt;3-&gt;{
    \draw [-&gt;,thick] (c.south) ..controls +(south:1.0em) and +(north:1.0em).. (clabel.north);
    }

    \visible&lt;5-&gt;{
    \draw [-&gt;,thick] ([yshift=1em]flabel.east) -- ([yshift=1em]clabel.west);
    \draw [&lt;-,thick] ([yshift=-1em]flabel.east) -- ([yshift=-1em]clabel.west) node [pos=0.5,above,yshift=0.3em] {\footnotesize{\textbf{����ִ��}}};
    }
    \end{tikzpicture}
    \end{center}
\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% the EM algorithm - training IBM model 1
\begin{frame}{������ģ��ѵ������ - EM�㷨}
\label{ibmtraining}

\begin{beamerboxesrounded}[upper=uppercolblue,lower=lowercolblue,shadow=true]{IBMģ��1��ѵ����EM�㷨��}
����: ƽ������$\{(s^{[1]},t^{[1]}),...,(s^{[N]},t^{[N]})\}$\\
���������$f(\cdot|\cdot)$������ֵ\\
1: \textbf{Function} \textsc{TrainItWithEM}($\{(s^{[1]},t^{[1]}),...,(s^{[N]},t^{[N]})\}$) \\
2: \ \ Initialize $f(\cdot|\cdot)$ \hspace{5em} $\rhd$ �����$f(\cdot|\cdot)$һ�����ȷֲ�\\
3: \ \ Loop until $f(\cdot|\cdot)$ converges\\
4: \ \ \ \ \textbf{foreach} $k = 1$ to $N$ \textbf{do}\\
5: \ \ \ \ \ \ \ \footnotesize{$c_{\mathbb{E}}(s_u|t_v;s^{[k]},t^{[k]}) = \sum\limits_{j=1}^{|s^{[k]}|} \delta(s_j,s_u) \sum\limits_{i=0}^{|t^{[k]}|} \delta(t_i,t_v) \cdot \frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}\normalsize{}\\
6: \ \ \ \ \textbf{foreach} $t_v$ appears at least one of $\{t^{[1]},...,t^{[N]}\}$ \textbf{do}\\
7: \ \ \ \ \ \ \ $\lambda_{t_v}^{'} = \sum_{s_u} \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[k]},t^{[k]})$\\
8: \ \ \ \ \ \ \ \textbf{foreach} $s_u$ appears at least one of $\{s^{[1]},...,s^{[N]}\}$ \textbf{do}\\
9: \ \ \ \ \ \ \ \ \ $f(s_u|t_v) = \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;s^{[k]},t^{[k]}) \cdot (\lambda_{t_v}^{'})^{-1}$\\
10: \ \textbf{return} $f(\cdot|\cdot)$
\end{beamerboxesrounded}
\vspace{-0.3em}
\begin{itemize}
\item&lt;2-&gt; \alert{\textbf{E-Step: ��4-5������$c_{\mathbb{E}}(\cdot)$��M-Step����6-9������$f(\cdot)$}}
\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% the EM algorithm - training IBM model 2
\begin{frame}{EM�㷨��IBMģ��2}
\begin{itemize}
\item EM�㷨����ֱ������ѵ��IBMģ��2(����ʡȥ�Ƶ�)
    \begin{itemize}
    \item ����1��$f(s_j|t_i)$ - ��IBMģ��1һ��
    \item ����2��$a(i|j,m,l)$ - �������
    \end{itemize}
\end{itemize}
\begin{enumerate}
\item \textbf{E-Step} (���ھ��$(s,t)$��$m=|s|,l=|t|$)
\begin{eqnarray}
c_{\mathbb{E}}(s_u|t_v;s,t) &amp; = &amp; \sum_{j=1}^{m} \sum_{i=0}^{l} \frac{f(s_u|t_v)a(i|j,m,l)\delta(s_j,s_u)\delta(t_i,t_v)}{\sum_{k=0}^{l} f(s_u|t_k)a(k|j,m,l)} \nonumber \\
c_{\mathbb{E}}(i|j,m,l;s,t) &amp; = &amp; \frac{f(s_j|t_i)a(i|j,m,l)}{\sum_{k=0}^{l} f(s_j|t_k)a(k|j,m,l)} \nonumber
\end{eqnarray}
\item \textbf{M-Step}
\begin{eqnarray}
f(s_u|t_v) &amp; = &amp; \frac{\sum_{k=0}^{K} c_{\mathbb{E}}(s_u|t_v;s^{[k]},t^{[k]})}{\sum_{s_u} \sum_{k=0}^{K} c_{\mathbb{E}}(s_u|t_v;s^{[k]},t^{[k]})}  \nonumber \\
a(i|j,m,l) &amp; = &amp; \frac{\sum_{k=0}^{K} c_{\mathbb{E}}(i|j;s^{[k]},t^{[k]})}{\sum_{i} \sum_{k=0}^{K} c_{\mathbb{E}}(i|j;s^{[k]},t^{[k]})} \nonumber
\end{eqnarray}
\end{enumerate}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% the remaining problems
\begin{frame}{IBMģ��1��2������������}
\begin{itemize}
\item �����ع�һ�£�IBMģ��1��2�����������������ݣ�
    \begin{enumerate}
    \item \textbf{��ģ} - {\color{ublue} \hyperref[ibmmodelingstart]{��\hspace{-0.2em}~\ref{ibmmodelingstart}ҳ}}$\sim${\color{ublue} \hyperref[ibmmodelingend]{��\hspace{-0.2em}~\ref{ibmmodelingend}ҳ}}
    \item \textbf{����} - {\color{ublue} \hyperref[ibmdecoding]{��\hspace{-0.2em}~\ref{ibmdecoding}ҳ}}
    \item \textbf{ѵ��} - {\color{ublue} \hyperref[ibmtraining]{��\hspace{-0.2em}~\ref{ibmtraining}ҳ}}
    \end{enumerate}
\item ���кܶ�ܶ�����û������
    \begin{itemize}
    \item ���룺{\color{ublue} \hyperref[simpledecodingalgorithm]{��\hspace{-0.2em}~\ref{simpledecodingalgorithm}ҳ}}�������Ľ����㷨�ֲܴ�
        \begin{itemize}
        \item ��Ҫ���Ǹ���ķ������ - beam search - ����½ڻ����
        \end{itemize}
    \item&lt;2-&gt; ѵ����EM�㷨�������ǣ�
        \begin{itemize}
        \item ��Ŀ�꺯��Ϊ��ͻ(��)������ʱ����������ֲ�����

        \begin{tikzpicture}

        \begin{scope}[scale=0.18]
        \draw[-,thick,ublue,domain=-4.2:3.5,samples=100] plot (\x,{ 1/14 * (\x + 4) * (\x + 1) * (\x - 1) * (\x - 3)});
        \draw[-&gt;] (-6,0) -- (5,0);
        \draw[-&gt;] (-5,-4) -- (-5,3);
        \node[anchor=west,inner sep=1pt] (label1) at (-3,3) {\tiny{�����ֵ��}};
        \draw[-&gt;] ([xshift=-3em]label1.south) -- (2.3,-0.5);
        \draw[-&gt;] ([xshift=-3em]label1.south) -- (-2.5,-1);
        \node [anchor=east] at (-6,0) {\tiny{\alert{��}ͻ(non-convex)����}};
        \end{scope}

        \begin{scope}[scale=0.18,xshift=10in]
        \draw[-,thick,ublue,domain=-4.2:3.5,samples=100] plot (\x,{ 1/7 * (\x + 1) * (\x + 1) - 2});
        \draw[-&gt;] (-6,0) -- (5,0);
        \draw[-&gt;] (-5,-4) -- (-5,3);
        \node[anchor=west,inner sep=1pt] (label1) at (-3,3) {\tiny{һ����ֵ��}};
        \draw[-&gt;] ([xshift=-3em]label1.south) -- ([xshift=-3em,yshift=-10em]label1.south);
        \node [anchor=east] at (-6,0) {\tiny{ͻ(convex)����}};
        \end{scope}

        \end{tikzpicture}

        �Ƚ����˵���IBMģ��1�е�����ֵ�㣬EM���Եõ�ȫ�����š�����ͨ�����IBMģ��1�Ĳ�����Ϊ�߽�ģ��(��ģ��2)�����룬��֤�����ѵ���ӽϺõĳ�ʼֵ��ʼ

        \item ����degenerate analysis������\\
        Googleһ��variational EM��bayesian inference
        \end{itemize}
    \end{itemize}
\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% papers and systems
\begin{frame}{GIZA++}
\begin{itemize}
\item IBMģ��3-5���Ǳ��̵̳��ص�
    \begin{itemize}
    \item ��ģ��Ը���һЩ����Ȼģ��Ҳ��ǿ��
    \item ѵ���ͽ��벻̫��
    \item �Ƽ�һЩ�Ķ�����
        {\footnotesize
        \begin{itemize}
        \item \textit{\textbf{A Program for Aligning Sentences in Bilingual Corpora}}.
              William A. Gale and Kenneth W. Church. 1993.
        \item \textit{\textbf{A systematic comparison of various statistical alignment models}}.
              Franz Och and Hermann Ney. 2003.
        \item \textit{\textbf{����IBMģ�͵�ͳ�ƻ������뼼�����о�}}.
              Фͩ. 2008. (˶ʿ����)
        \end{itemize}
        }
    \end{itemize}
\item&lt;2-&gt; \textbf{���ò����GIZA++}��ʵ����IBMģ��+HMMģ�ͣ����ڹ㷺����˫��ƽ�����ϵ��Զ��ʶ���
    \begin{itemize}
    \item ��ȫ��Դ\\
          \url{http://code.google.com/p/giza-pp/}
    \item NiuTrans��Ҳ��ʹ�÷�ʽ�Ľ���\\
          \url{http://www.nlplab.com/NiuPlan/NiuTrans.YourData.ch.html}
    \end{itemize}
\end{itemize}
\end{frame}

%%%------------------------------------------------------------------------------------------------------------
%%% last slide
\begin{frame}{����������д�����~}

\vspace{2em}

\begin{center}
\textbf{��������д�˺ܶ࣬���ǻ�ûŪ�� :(}\\
\textbf{����(���ܹ�����)���ϵ����Զ���������� :)}

\vspace{3em}

\begin{tikzpicture}

{\LARGE
\node [anchor=west,inner sep=2pt,ublue] (s1) at (0,0) {лл};
\node [anchor=west,inner sep=2pt,ublue] (s2) at ([xshift=1.0em]s1.east) {���};
\node [anchor=north,inner sep=2pt,ublue] (t1) at ([yshift=-1.7em]s1.center) {thank};
\node [anchor=north,inner sep=2pt,ublue] (t2) at ([yshift=-1.95em]s2.center) {you};
\node [anchor=west,inner sep=2pt,ublue] (t3) at ([xshift=1.5em,yshift=0.2em]t2.east) {all};

\draw [-,thick,ublue] ([yshift=-0.1em]s1.south) -- (t1.north);
\draw [-,thick,ublue] ([yshift=-0.1em]s2.south) -- ([yshift=0.2em]t2.north);
\draw [-,thick,ublue] ([yshift=-0.1em]s2.south) -- (t3.north);

}

\end{tikzpicture}

\end{center}

\end{frame}

\end{CJK}
\end{document}
</pre></body></html>