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7.5 Probabilistic Context-Free Grammars

• Category C$B$KBP$7(B,$B$I$N(B rule $B$,$I$l$@$1E,MQ$5$l$?$+$N3NN((B
  
  $$\begin{eqnarray*}C &\rightarrow& R_1 \\ C &\rightarrow& R_2 \\ &\vdots& \\ C &\rightarrow& R_m \end{eqnarray*}$$
  
  $B$N$H$-(B rule R_j $B$,;HMQ$5$l$k3NN((B PROB(R_j|C) $B$O(B
  
  $$\begin{eqnarray*}PROB(R_j\vert C) \simeq \frac{COUNT(\char93 \ times\ R_j\ used)} {\sum_{i=1}^m COUNT(\char93 \ times\ R_i\ userd)} \end{eqnarray*}$$
  
  Figure 7.17
• Viterbi algorithm $B$r3HD%$9$l$P3NN($O4JC1$K5a$^$k(B
• $BLdBjE@(B
  • $B3F(B rule $B$N@8@.3NN($rFHN)$H$7$F$$$k(B
  • R_j $B$N9=@.AG$,(B, $B$=$N>e$G2?$NMWAG$K$J$k$N$+9MN8$5$l$J$$(B
  • NP $\rightarrow$ ART N $B$G(B, NP $B$,(B subject $B$K$J$k$+(B, verb of object $B$K$J$k$+(B object of a preposition$B$K$J$k$+(B?
  • subject $BCf$N(B NP $B$O(B, NP $B$h$j(B pronouns $B$K$J$j$d$9$$(B
• inside probability
  • category C $B$,C18lNs(B $w_i,w_{i+1},\cdots,w_j$$B$KE83+$5$l$k3NN((B
    
    $$\begin{eqnarray*}PROB(w_i,w_{i+1},\cdots,w_j\vert C) \end{eqnarray*}$$
    
    
  • Figure 7.18,7.19
    
    $$\begin{eqnarray*}PROB(a\ flower\ blooms\vert S) \end{eqnarray*}$$
    
    $B$r5a$a$F$_$h$&(B (buttom up)
    
    PROB(flower|N),PROB(a|ART),PROB(a|N) $\rightarrow$ Figure 7.6
    
    $$\begin{eqnarray*}PROB(a\ flower\vert NP) &&= \\ && PROB(Rule\ 8\vert NP) \cdot... ...cdot 0.36 \cdot 0.6 + 0.9 \cdot + 0.01 \cdot 0.6 \\ &&= 0.012 \end{eqnarray*}$$
    
    
    
    $$\begin{eqnarray*}PROB(a\ flower\ blooms\vert S) &&= \\ && PROB(Rule\ 1) \cdot ... ...ert S) \cdot PROB(a\vert NP) \cdot PROB(flower\ blooms\vert VP) \end{eqnarray*}$$
    
    
  $B$7$+$7(B sentence $B$,$"$kFCDj$NC18lNs$KE83+$5$l$k3NN($r5a$a$F$b0UL#(B $B$,$J$$(B, $B$`$7$mC18lNs$,M?$($i$l$?$H$-$K$I$N$h$&$J(B rule $B$,E,MQ$5$l(B $B$k$+$,=EMW(B
• $B$"$kFCDj$N(B parse tree$B$,E,MQ$5$l$k3NN((B
  • $BC18lNs(B $w_i,w_{i+1},\cdots,w_j$$B$KBP$7$F(B, $B$"$k(B parse tree $B$,F@$i$l(B $B$k3NN($r5a$a$k(B
    
    $$\begin{eqnarray*}E_{category = C} \rightarrow E_1\ E_2 \cdots E_n \end{eqnarray*}$$
    
    $B$H$$$&(B rule i $B$,E,MQ$5$l$k3NN($O(B
    
    $$\begin{eqnarray*}PROB(E) = PROB(Rule\ i\vert C) \cdot PROB(E_i) \cdot \cdots \cdot PROB(E_n) \end{eqnarray*}$$
    
    $B$H$J$k(B
  • PROB(E) $B$,(B lexical category ($BIJ;l(B) $B$K$J$C$?>l9g$O(B, 7-4 $B$G=R$Y$?(B Lexical Probabilites $B$r;HMQ$9$k(B
    context-independent $B$J3NN((B (Figure 7.13)$B$G$O(B..
    
    $$\begin{eqnarray*}PROB(flower/N) \rightarrow 0.78\\ PROB(flower/V) \rightarrow 0.22 \end{eqnarray*}$$
    
    $B$G$"$j(B V $B$r$_$?$9>l9g9M$($i$l$k(B,$B$7$+$7(B context-dependent (Figure 7.16) $B$D$^$j(B, $PROB(flower/V\vert a\ flower)$ $B$r5a$a$k$H(B flower/V $B$K$J$k2DG=@-$O6K$a$FDc$$(B (a $\rightarrow$ N, flower $\rightarrow$ V)
  • $B6qBNNc(B
    ``a flower'' $B$KBP$7$F(B, context-dependent $B$J3NN($r;H$C$F(B Chart Parsing $B$r9T$C$?Nc(B $\rightarrow$ Figure 7.20
  • $B$7$+$7$3$l$G$b(B50% $BDxEY$N@:EY$7$+F@$i$l$J$$(B $\rightarrow$ $B3F(B rule $B$rFHN)$J$b$N$H$7$F2>Dj$7$F$$$k$?$a(B
    
    ($B6qBNNc(B) Figure 7.21
    
    $$\begin{eqnarray*}VP &\rightarrow& V\ NP\ PP\ (0.22) \\ V &\rightarrow& V\ NP\ (0.39) \\ NP &\rightarrow& NP\ PP\ (0.24) \end{eqnarray*}$$
    
    VP $B$,(B (V,NP,PP) $B$KE83+$5$l$k3NN($O05E]E*$K(B Figure 7.21 $B$N:8$NLZ$N2DG=@-$,9b$$(B,
    courps $BCf$K(B 23 $BNc$"$k(B NP $\rightarrow$ NP PP $B$O(B $B$9$Y$F8m$C$F2r@O$5$l$F$7$^$&(B
    
    84$B$NJ8$G(B $B3NN($r;H$o$J$+$C$?>l9g(B 1/3 $B$,@52r(B
    $B3NN($r;H$C$?>l9g(B 1/2 ($B$?$@$7(B closed test), open test $B$K$7$?$i$b$C$H0-$/$J$k(B
    
    context $B$K0MB8$7$?3NN($r;H$C$F(B parsing $B$r9T$&I,MW$"$j(B