The dissertation of Andreas Stolcke is approved: University of ...

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¸0 ¸ 29¸¸ 99W9 [;t£ ? 1 ?1u 6¬t [;t£9W9 [7t£ ? u1u 6/t [_t£9W99š9%££9W99š9%£9š9%£1 Ngt??1 0 u 3 [7t 6,???,?,???òu óÆ1 tã6 1 t1 u t?utK6?ŠuCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 140(a)9tró(b)State set 00 ¸ £j9predicted11 tã6 1 tò109 ¸¸State set 1scanned0911u¸completed09B£ tpredicted9%£9 tu [;tK6/t u1u [;tK6 u¸State set 2scanned19¸¸ 19 091u 2completedu tB£2 [7tã6 u 21uu [ u 6/t uu[7tã6¬t u t u [7tã6/t 2u1u [;t 2u 6¬t u 2 u [_t 2u 6Àt 2u 20 ¸predicted¸¸19¸¸ 09 19 1 11 6 ,1 NgtE0 u 2tt£ 1 NÜt1 0 u 3 uu 2 Ngt u 2 0%[7t ?1u 6, u 2 Ngt u 2 0%[_t ?¸State set 3scanned2929 ¸¸completed29B£NgtE0 u 2t1NgtE0 u 3 u [;tK6/t u1,2 [_t96¬t u 2t u [7tã6/t 2u1u 2 [;t 2u 6¬t u 3 u [_t 2u 6Àt 2u 29%£9,utu 2 [;t§N u [7t 2u 6 2t 2u 2t 2u 2 [;t§NÜt u [7t 2u 6 2t 3u 2tt 1u [ 2t 3u 2 6 2t 2u 3 u [ 2t 3u 2 6 2t 3u 39%£99%£99%£ 1 [ 2t 3u 2 6 2t 3u 21 [ 2t 3u 2 6 2t 3u 2¸¸¸ ¸19 09Table 6.2: Earley chart as constructed during the parse of .The grammar is depicted in (a). The two columns to the right in (b) list the forward and innerprobabilities, respectively, for each state. In both Æ andŠ columns, the [ separates old factorsfrom new ones (as per equations 6.1, 6.2, 6.3). Addition indicates multiple derivations of thesame state.
A prerequisite for this approach is to precompute, for all nonterminals ) , the probability that )”””Ó”¸=1=2òò””CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 141section summarizes the necessary modifications to process null productions correctly, using the previousdescription as a baseline. Our treatment of null productions follows the (non-probabilistic) formulation ofGraham et al. (1980), rather than the original one in Earley (1970).6.4.7.1 ComputingA-expansion probabilitiesThe main problem with null productions is that they allow multiple prediction-completion cyclesin between the scanning steps (since null productions do not have to be matched against one or more inputsymbols). Our strategy will be to collapse all predictions and completions due to chains of null productionsinto the regular prediction and completion steps, not unlike the way recursive predictions/completions werehandled in Section 6.4.5.expands to the empty string. Note that this is another recursive problem, ) sinceproduction, but expand to nonterminal= some that does.Computation ) of for ) allFor +-, each , let be an abbreviation for +-, C«A0)Ó )itself may not have a nullcan be cast as a system of non-linear equations, as follows.. By way of example, if ) CEA0 has productions) ¸ Atò1ót 2ó.¸=3=4=5t 3óThe semantics of context-free rules imply that ) can only expand toAif all of the RHSs in one of ) ’sproductions expands toA.Translating to probabilities, we obtain the equation6/t 1 Ngt 2Ó‘1Ó‘2Ngt 3Ó‘3Ó‘4Ó‘5N|£££In other words, each production contributes a term in which the rule probability is multiplied by the productof the Ó variables corresponding to the RHS nonterminals, unless the RHS contains a terminal (in which casethe production contributes nothing Ó to because it cannot possibly lead toA).The resulting non-linear system can be solved by iterative approximation. Each Ó variableinitialized to +-, ) ¸A0 , and then repeatedly updated by substituting in the equation right-hand sides, untilisincreasing and bounded above by the true values +9, )the desired level of accuracy is attained. Convergence is guaranteed since Ó the values are monotonically1. For grammars without cyclic dependencies CEA0ŒVamongA-producing nonterminals this procedure degenerates to simple backward substitution. Obviously thesystem has to be solved only once for each grammar.The Ó probability can be seen as the precomputed inner probability of an expansion ) of to theempty string, i.e., it sums the probabilities of all Earley paths that ) deriveAfrom . This is the justificationfor the way these probabilities can be used in modified prediction and completion steps, described next.
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The dissertation of Andreas Stolcke
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Bayesian Learning of Probabilistic
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iAcknowledgmentsLife and work in Be
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iiiContentsList of FiguresList of T
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CONTENTSv4.5.4 Summary and Discussi
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CHAPTER 1. INTRODUCTION 2Instance-b
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CHAPTER 1. INTRODUCTION 4A.0.830.33
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CHAPTER 1. INTRODUCTION 6the ¨ 0 l
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..1 £££1; 450,1 £££1; 450CHAP
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VU=@U@@=U===UCHAPTER 2. FOUNDATIONS
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,,vv,v,v,,directly. However, note t
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4@@@@-@b@6@˜--@@@0@@@@@CHAPTER 2.
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6tt,u ·¥¸¹u ,10ºtu ,2 10Yt ¸
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CHAPTER 2. FOUNDATIONS 18As more da
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CHAPTER 2. FOUNDATIONS 20Global mod
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CHAPTER 2. FOUNDATIONS 22¡An expli
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ÊS==66@N,ÆÆ=NÆ00ÆÊ=S=N0Æ=#@0
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666CHAPTER 2. FOUNDATIONS 262.5.7 P
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It, u¦¸¹u Ù 0w6¬tt,_, u Ù 0
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, uu!¸¹u Ù 0c6,u ,ÔÔ0 ö1 ö1
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CHAPTER 3. HIDDEN MARKOV MODELS 32T
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CHAPTER 3. HIDDEN MARKOV MODELS 34R
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4ÿ= ê•4TÃE0&Ò¢¡? •ç1 Lht
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6666ò U1ò +9,9. 4 20+-,¡ . 4 10C
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2. For each candidate"I!computeLet"
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6\“ç%&ät\“ç tè ä, u¦¸¹u
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, u1 ¸¼u Ù 0 and , u3 ¸¹u Ù 0
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CHAPTER 3. HIDDEN MARKOV MODELS 46l
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CHAPTER 3. HIDDEN MARKOV MODELS 48c
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CHAPTER 3. HIDDEN MARKOV MODELS 50d
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CHAPTER 3. HIDDEN MARKOV MODELS 520
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correlation between initial and fin
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,CHAPTER 3. HIDDEN MARKOV MODELS 56
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,CHAPTER 3. HIDDEN MARKOV MODELS 60
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CHAPTER 3. HIDDEN MARKOV MODELS 62b
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CHAPTER 3. HIDDEN MARKOV MODELS 64t
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CHAPTER 3. HIDDEN MARKOV MODELS 66t
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CHAPTER 3. HIDDEN MARKOV MODELS 68s
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CHAPTER 3. HIDDEN MARKOV MODELS 74b
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domain. 3 In short, we will leave o
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,,,,,£CHAPTER 4. STOCHASTIC CONTEX
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9 ¸)Ô ¸ 9 ¸Ô 1 2 £££;,ÔÔC
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¸= ¸= ¸.1.2¸¸) 1_) 20&6#=,,,,
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CHAPTER 4. STOCHASTIC CONTEXT-FREE
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Page 102 and 103: CHAPTER 4. STOCHASTIC CONTEXT-FREE
Page 104 and 105: ==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
Page 106 and 107: ,= ===I¸theybÜ„thiscg„\\ ¸¸
Page 116 and 117: 104Chapter 5Probabilistic Attribute
Page 118 and 119: ,1makingandCHAPTER 5. PROBABILISTIC
Page 120 and 121: CHAPTER 5. PROBABILISTIC ATTRIBUTE
Page 134 and 135: 122Chapter 6Efficient parsing with
Page 136 and 137: 1z1CHAPTER 6. EFFICIENT PARSING WIT
Page 138 and 139: and each state in set ( -ÏH ):6)
Page 140 and 141: NPDetVTVI P CHAPTER 6. EFFICIENT PA
Page 142 and 143: CHAPTER 6. EFFICIENT PARSING WITH S
Page 144 and 145: ) In particular, the string probabi
Page 146 and 147: H : d=:6) ¸ÆÆÆ:6) ¸ .V=i£Ù )
Page 148 and 149: ©) 6#=©,©,©,,ÆNNö,²NL++and>+
Page 150 and 151: ) The probabilistic unit-production
Page 154 and 155: The forward and inner probabilities
Page 156 and 157: 9 itself²²NN++9 ¸0ÌLL++??1£,=C
Page 158 and 159: ,,by nonterminals. Multiplying this
Page 160 and 161: 6CHAPTER 6. EFFICIENT PARSING WITH
Page 162 and 163: description. Again, we ignore this
Page 164 and 165: for all pairs of states d =¸+= Š
Page 166 and 167: ,9 ¸0 : 0¸ £j9¸ A )z9£CHAPTER
Page 170 and 171: are then summed over all nontermina
Page 172 and 173: +CHAPTER 6. EFFICIENT PARSING WITH
Page 174 and 175: 1CHAPTER 6. EFFICIENT PARSING WITH
Page 178 and 179: = ¸Let t,t) ¸ .V=i£,t,t,6666yyyy
Page 180 and 181: 168Chapter 7-grams from Stochastic
Page 182 and 183: CHAPTER 7. -GRAMS FROM STOCHASTIC
Page 184 and 185: )ÅÆÅÅÅÆÅÅÅÆÅÅÅÆÅÅÅ
Page 186 and 187: -grams CCHAPTER 7. -GRAMS FROM ST
Page 188 and 189: ,?Ó,tÌ?L A0,I 1N I N A A 2 A 3 N
Page 190 and 191: ,,CHAPTER 7. -GRAMS FROM STOCHASTI
Page 192 and 193: Consider the following problem: sta
Page 194 and 195: CHAPTER 8. FUTURE DIRECTIONS 1828.2
Page 196 and 197: 184BibliographyAHO, ALFRED V., RAVI
Page 198 and 199: BIBLIOGRAPHY 186DAGAN, IDO, FERNAND
Page 200 and 201: BIBLIOGRAPHY 188——, & ——. 1
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BIBLIOGRAPHY 190QUINLAN, J. ROSS, &
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BIBLIOGRAPHY 192WALLACE, C. S., & P
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