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

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description. Again, we ignore this technicality in our terminology. The term is motivated by the fact that´,,1,661,00derives@‰yy,,isinCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 150Definition 6.9 Given a string 1 , .1the probabilities of all paths that.6{G , the outer probability´@_,6)¸.—£ ˜M0 of an Earley state is the sum of¡start with the initial state,¡generate the prefix 1 0 £££1 ,16?¡pass through6)¸ £j@$˜ , for some@,generate the suffix 1 @ £££1ED ?¡1 starting with state6)¸@>£ ˜ ,end in the final state.¡Outer probabilities complement inner probabilities in that they refer to precisely to those partsof complete paths generating not covered by the corresponding inner probabilityŠ@7,6)¸ .—£ ˜M0 . A1potentially confusing aspect of this definition is that the choice of the production ) ¸outer probability associated with state6)¸a.of the RHS: all states sharing the ) same",˜ and will identical´@have ..>˜ is not part of the.M£ ˜ . In fact, the definition makes no reference to the first part.—£ ˜M0 is the probability that an Earley parser operating as a string generatoryields the prefix 1 0å…å…å6?¸Intuitively,´@_,6)1 and theD ?å…å…å suffix 1, while passing state6)¸throughis independent of.). As was the case for 1 forward is actually an expectation of the number@ probabilities,´.M£ ˜ at position H (whichof such states in set H , as unit production cycles can lead to paths that have more than one state fitting thisreduces to the ‘outer probability’ of )as defined in Baker (1979) if the dot is in final position.6.5.2.1 Computing expected production countsBefore going into the details of computing outer probabilities we briefly describe their use inobtaining the expected rule counts needed for the E-step in grammar estimation.Let ¸ .%. )string . Alternatively, 1complete Earley path generating . Let production 1 ¸.along a path‰.)denote the expected number of uses of production ) ¸ 0) ¸ .w. 0 is the expected number of times that ) ¸ . .) ¸the derivation ofused for prediction in a.%.‰ë0 be the number of occurrences of predicted states with0_.%.‰ë0) ¸.w.1) ¸y 06derives+-,‰K.9 1C 1 ‰1C10_.%.‰ë0) ¸1+-,9 1 +-,‰ ?91)z@ C1@C?0å…å…å of all paths passing through H : @ ) ¸£.. Inner and outer probabilities have been defined such that this1)z@+-,9 0å…å…å@A?C10F£C1C1Summation is over all predicted states using ) ¸.. +-,9 :^0 is the sum of the probabilities”' å/+-,9C1C1
,H :6) ¸ .}=Œ£H :6) ¸ .t>£˜˜ò´&)Šó 6QCò´&)Šó 6QC0@y´Ù += ŠÙ…Ù´: d§=¸¸6) ::6) ¸óò´Ù…Ùò´Ùñóò´Ùñóò´ÙñóCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 151expected count is obtained precisely as the product of the corresponding ofŠ@usage count for a rule can be computed as. Thus, the expectedand´@1@@) ¸.%.1) ¸ £j.E0Š@The sum can be computed after completing both forward and backward passes (or during the backward passitself) by scanning the chart for predicted states.+-,9 :^0%6C1) ¸ £j.B0 £”' å/´@ ,,6.5.2.2 Computing outer probabilitiesThe outer probabilities are computed by tracing the complete paths from the final state to the startstate, in a single backward pass over the Earley chart. Only completion and scanning steps need to be tracedback. Reverse scanning leaves outer probabilities unchanged (similar to inner and forward probabilities inthe forward pass), so the only operation of concern is reverse completion.We describe reverse transitions using the same notation as for their forward counterparts, exceptthat each state is written with its outer and inner probabilities.Reverse completion: d =@$£.—£=-˜for all pairs of states¸ d=.—£=¤˜ in the chart. ThenµH@>£ and6)¸The inner probabilityŠis not used.Rationale. Relative to´ÁÙ ,´´Ù…Ù += ŠÙ´missing the probability of expanding=, which is filled in fromŠÍ˜
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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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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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2. For each candidate"I!computeLet"
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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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==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
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,= ===I¸theybÜ„thiscg„\\ ¸¸
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Page 112 and 113: CHAPTER 4. STOCHASTIC CONTEXT-FREE
Page 114 and 115: CHAPTER 4. STOCHASTIC CONTEXT-FREE
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 152 and 153: ¸0 ¸ 29¸¸ 99W9 [;t£ ? 1 ?1u 6
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 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
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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
Page 202 and 203: BIBLIOGRAPHY 190QUINLAN, J. ROSS, &
Page 204: BIBLIOGRAPHY 192WALLACE, C. S., & P
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