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

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H : d=:6) ¸ÆÆÆ:6) ¸ .V=i£Ù )ŠÙÙ ŠÙCHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 134of choosing production=¸@. The valueŠEÙ is just a special case of the definition.Rationale. Æ Ù is the sum of all path probabilities leading up to6)¸.M£=§˜ , times the probabilityScanning (probabilistic)for all states with terminal matching input at position H . Then.—£ f˜)Šó 6QC.BE£ ˜óH :6) ¸1 :6) ¸ò Æò…ÆHBNÙ 66 ŠÙRationale. Scanning does not involve any new choices since the terminal was already selected asŠpart of the production during prediction. 10Completion (probabilistic)ò…Æ@$£Ù…ÙY)Š>Ù…Ùñóò…Æ¸ò…Æ˜ó.—£=¤˜)ŠóŽ 6QC HThenÙ += ÆŠÙ…Ù (6.1)+= Š}ŠÙ…Ù (6.2) ŠÙNote that Ù is not used.Rationale. To update the old forward/inner probabilities Æ andŠ to Æ Ù andŠ>Ù , respectively, theÆprobabilities of all expanding=¸paths @have to be factored in. These are exactly the paths summarizedby the probabilityŠBÙ…Ù inner .6.4.5 Coping with recursionThe standard Earley algorithm, together with the probability computations described in the previoussection would be sufficient if it weren’t for the problem of recursion in the prediction and completion steps.The non-probabilistic Earley algorithm can stop recursing as soon as all predictions/completionsyield states already contained in the current state set. For the computation of probabilities, however, thiswould mean truncating the probabilities resulting from the repeated summing of contributions.10 In different parsing scenarios the scanning step may well modify probabilities. For example, if the input symbols themselves haveattached likelihoods these can be integrated by multiplying them onto and when a symbol is scanned. That way it is possible toperform efficient Earley parsing with integrated joint probability computation directly on weighted lattices of input symbols.
ÆÆb) The probabilistic left-corner relation 11 +ÆÆ¸¸ 99 0¸ £9£ òu óò£…06 t§N u t§N u 2t§N|£££6¬t) ¸ =.+ø,??iff?CHAPTER 6. EFFICIENT PARSING WITH STOCHASTIC CONTEXT-FREE GRAMMARS 1356.4.5.1 Prediction loopsAs an example, consider the following simple left-recursive SCFG.tówhere u 6 1 L§t . Non-probabilistically, the prediction loop at position 0 would stop after producing the states9%09 ¸09 ¸This would leave the forward probabilities at£9% £,0£…06 t,0£j9%06 u corresponding to just two out of an infinity of possible paths. The correct forward probabilities are obtainedas a sum of infinitely many terms, accounting for all possible paths of length 1.¸¸ 09 09,0¸091 L u 016 1,0u N u 2N u 3N|£££6 u ,1 L u 0£j9%0616¬t¸ 1uIn these eacht sums corresponds to a choice of the first production, each to a choice of the second production.u09If we didn’t care about finite computation the resulting geometric series could be computed by letting theprediction loop (and hence the summation) continue indefinitely.Fortunately, all repeated prediction steps, including those due to left-recursion in the productions,can be collapsed into a single, modified prediction step, and the corresponding sums computed in closed form.For this purpose we need a probabilistic version of the well-known parsing concept of a left corner, which isalso at the heart of the prefix probability algorithm of Jelinek & Lafferty (1991).Definition 6.5 The following definitions are relative to a given SCFG{.a) Two nonterminals ) and= are said to be in a left-corner relation ) ¸ =for )that has a RHS starting with=,there exists a productiondefined as the total probability of choosing a production for )that has= as a left corner:’6£is the matrix of probabilities +-, ) ¸ L=ë0 ,,{§0+-,+9,) ¸ y ”'’‘/“ =ë0&6) ¸ =“.B0Â£11 If a probabilistic relation”is replaced by its version”œ set-theoretic , i.e.,used here reduce to their traditional discrete counterparts; hence the choice of terminology.§¨+•‡Q–—”œthen the closure operationsiff”io§¨˜•‡š0,
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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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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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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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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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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 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 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
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Page 192 and 193: Consider the following problem: sta
Page 194 and 195: 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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