figure-em-algorithm-flow-chart.tex

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\node [anchor=north west] (line1) at (0,0) {\small\sffamily\bfseries{IBM模型1的训练（EM算法）}};
\node [anchor=north west] (line2) at ([yshift=-0.3em]line1.south west) {输入: 平行语料${(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[N]},\mathbf{t}^{[N]})}$};
\node [anchor=north west] (line3) at ([yshift=-0.1em]line2.south west) {输出: 参数$f(\cdot|\cdot)$的最优值};
\node [anchor=north west] (line4) at ([yshift=-0.1em]line3.south west) {1: \textbf{Function} \textsc{EM}($\{(\mathbf{s}^{[1]},\mathbf{t}^{[1]}),...,(\mathbf{s}^{[N]},\mathbf{t}^{[N]})\}$) };
\node [anchor=north west] (line5) at ([yshift=-0.1em]line4.south west) {2: \ \ Initialize $f(\cdot|\cdot)$ \hspace{5em} $\rhd$ 比如给$f(\cdot|\cdot)$一个均匀分布};
\node [anchor=north west] (line6) at ([yshift=-0.1em]line5.south west) {3: \ \ Loop until $f(\cdot|\cdot)$ converges};
\node [anchor=north west] (line7) at ([yshift=-0.1em]line6.south west) {4: \ \ \ \ \textbf{foreach} $k = 1$ to $N$ \textbf{do}};
\node [anchor=north west] (line8) at ([yshift=-0.1em]line7.south west) {5: \ \ \ \ \ \ \ \footnotesize{$c_{\mathbb{E}}(\mathbf{s}_u|\mathbf{t}_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]}) = \sum\limits_{j=1}^{|\mathbf{s}^{[k]}|} \delta(s_j,s_u) \sum\limits_{i=0}^{|\mathbf{t}^{[k]}|} \delta(t_i,t_v) \cdot \frac{f(s_u|t_v)}{\sum_{i=0}^{l}f(s_u|t_i)}$}\normalsize{}};
\node [anchor=north west] (line9) at ([yshift=-0.1em]line8.south west) {6: \ \ \ \ \textbf{foreach} $t_v$ appears at least one of $\{\mathbf{t}^{[1]},...,\mathbf{t}^{[N]}\}$ \textbf{do}};
\node [anchor=north west] (line10) at ([yshift=-0.1em]line9.south west) {7: \ \ \ \ \ \ \ $\lambda_{t_v}^{'} = \sum_{s_u} \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]})$};
\node [anchor=north west] (line11) at ([yshift=-0.1em]line10.south west) {8: \ \ \ \ \ \ \ \textbf{foreach} $s_u$ appears at least one of $\{\mathbf{s}^{[1]},...,\mathbf{s}^{[N]}\}$ \textbf{do}};
\node [anchor=north west] (line12) at ([yshift=-0.1em]line11.south west) {9: \ \ \ \ \ \ \ \ \ $f(s_u|t_v) = \sum_{k=1}^{N} c_{\mathbb{E}}(s_u|t_v;\mathbf{s}^{[k]},\mathbf{t}^{[k]}) \cdot (\lambda_{t_v}^{'})^{-1}$};
\node [anchor=north west] (line13) at ([yshift=-0.1em]line12.south west) {10: \ \textbf{return} $f(\cdot|\cdot)$};

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