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

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NPDetVTVI P CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 128(a)SVPDet aNPNN circle€square€triangleVPNPVT touchesVPPPVI isPPNPP above€below(b)¸ £ a£ ¸ circle£ touches£ ¸ a£ ¸ triangle£¸a circle touches a square0 S scanned scanned scanned scanned scannedpredicted 0Det 1N 2VT 3Det 4N¸ £ 09 £ ¸ Det£ ¸ N£ ¸ ¸ VT£ Det£ ¸ N££ ¸ NP£NP£ £ ¸ £ £ VP£ ¸NP £ £ ¸ £ ¸ £ S£¸NP VP completed completed completed completed completed0NP Det N 0 NP N 0NP Det 2VP NP 3NP N 4NP Det0Det a predicted 0S VP predicted predicted 3VP VT1N circle predicted 3NP Det N 5N circle 0S NP1N square 2VP VT 3Det a 4N square 0¸ ¸ £ ¸ £ ££1N triangle 2VP VI PP 4N triangle2VT touches2VI £ isState set 0 1 2 3 4 5¸Table 6.1: Example of non-probabilistic Earley-parsing.(a) Example grammar for a tiny fragment of English. (b) Earley parser processing the sentencea circle touches a triangle.
£££§C11y‚“sê„ƒ+-,9 0&61CCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 129where@1?@2£££?@6are string of terminals and/or nonterminals, ) ¸.is production of{, and@2 isderived from@1 by replacing one occurrence of )with..derivations ) C1producing from ) . 6 1 Cb) The string probability +-, ) 0 (of 1 given ) ) is the sum of the probabilities of all left-most1c) The sentence 1probability C +9,9(of given{) is the string probability given the symbol9 start 0 of{. By definition, this is also the 1 probability assigned to by the grammar{.1 .{§0 +-,21d) The prefix probability C¬+-,9having 1 as a prefix,0 (of 1 given{) is the sum of the probabilities of all sentence strings C+-,9÷s0Â£(In particular, 1). CTzA0w6 +9,9In the following, we assume that the probabilities in a SCFG are proper and consistent as definedin Booth & Thompson (1973), and that the grammar contains no useless nonterminals (ones that can neverappear in a derivation). These restrictions ensure that all nonterminals define probability measures overstrings, i.e., +-, )C10 is a proper distribution over 1 for all ) . Formal definitions of these conditions aregiven in Section 6.4.8.6.4.2 Earley paths and their probabilitiesIn order to define the probabilities associated with parser operation on a SCFG, we need the conceptof a path, or partial derivation, executed by the Earley parser.Definition 6.2a) An (unconstrained) Earley path, or simply path, is a sequence of Earley states linkedby prediction, scanning, or completion. For the purpose of this definition, we allow scanning to operatein ‘generation mode,’ i.e., all states with terminals to the right of the dot can be scanned, not just thosematching the input. (For completed states, the predecessor state is defined to be the complete state fromthe same state set contributing to the completion.)b) A path is said to be constrained by (or generate) a string 1 if in all scanned states the terminalsimmediately to the left of the dot, in sequence, form the string 1 .c) A path is complete if the last state on it matches the first, except that the dot has moved to the end ofthe RHS.d) We say that a path starts with nonterminal ) if the first state on it is a predicted state with ) on theLHS.6 In a left-most derivation each step replaces the nonterminal furthest to the left in the partially expandedstring. The order of expansionis actually irrelevant for this definition, due to the multiplicative combination of production probabilities. We restrict summation toleft-most derivations to avoid counting duplicates, and because left-most derivations will play an important role later.
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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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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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CHAPTER 3. HIDDEN MARKOV MODELS 34R
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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
Page 90 and 91: ,,,,,£CHAPTER 4. STOCHASTIC CONTEX
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Page 104 and 105: ==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
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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 142 and 143: CHAPTER 6. EFFICIENT PARSING WITH S
Page 144 and 145: ) In particular, the string probabi
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Page 148 and 149: ©) 6#=©,©,©,,ÆNNö,²NL++and>+
Page 150 and 151: ) The probabilistic unit-production
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Page 154 and 155: The forward and inner probabilities
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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
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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
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Page 186 and 187: -grams CCHAPTER 7. -GRAMS FROM ST
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,,CHAPTER 7. -GRAMS FROM STOCHASTI
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Consider the following problem: sta
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CHAPTER 8. FUTURE DIRECTIONS 1828.2
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184BibliographyAHO, ALFRED V., RAVI
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BIBLIOGRAPHY 186DAGAN, IDO, FERNAND
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BIBLIOGRAPHY 188——, & ——. 1
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BIBLIOGRAPHY 190QUINLAN, J. ROSS, &
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BIBLIOGRAPHY 192WALLACE, C. S., & P
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