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

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,,CHAPTER 7. -GRAMS FROM STOCHASTIC CONTEXT-FREE GRAMMARS 178Word error (%)Bigram estimated from raw data 33.7Bigram computed from SCFG 32.9by Monte-Carlo sampling 33.3Bigram from SCFG plus data 29.6SCFG 29.6Mixture Bigram-SCFG 28.8Table 7.2: Speech recognition accuracy using various language models.the other hand, is a standard weighted mixture of two distinct submodels, as described in Section 2.3.1.The experiments therefore support the argument made earlier that more sophisticated languagemodels, even if far from perfect, can ¢ improve -gram estimates obtained directly from sample data. Wealso see that the bulk of the improvement does not come from using a SCFG alone, but from smoothing thebigram statistics through the constraints imposed by the SCFG (possibly combined with a mixture of the twolanguage models).7.7 SummaryWe have described an algorithm to compute in closed form the distribution ¢ of -grams for aprobabilistic language given by a stochastic context-free grammar. The algorithm is based on computingsubstring expectations, which can be expressed as systems of linear equations derived from the grammar.Listed below are the steps of the ¢ complete -gram-from-SCFG computation. For concreteness we give theversion specific to bigrams (¢ 6 2).1. Compute the prefix (left-corner) and suffix (right-corner) probabilities for each (nonterminal,word)pair.2. Compute the coefficient matrix and right-hand sides for the systems of linear equations, as per equations(7.4) and (7.5).3. LU decompose the coefficient matrix.4. Compute the unigram expectations for each word in the grammar, by solving the LU system for theunigram right-hand sides computed in step 2.5. Compute the bigram expectations for each word pair by solving the LU system for the bigram right-handsides computed in step 2.unigram expectation ( 1.9%0 .6. Compute each bigram probability +-, ( 2. ( 10 , by dividing the bigram expectation ( 1( 2.9%0 by the
¸ > ) ¸are all productions with LHS ) , the entry indexed by ) in the right-hand side vector would be +9, ) ¸¸ )¸ ˜.CHAPTER 7. -GRAMS FROM STOCHASTIC CONTEXT-FREE GRAMMARS 179The algorithm has been found practical in the context of a medium-scale speech understandingsystem, were it gave improved estimates for a bigram language model, based on a hand-written SCFG andvery small amounts of available training data.¢ Deriving -gram probabilities from more sophisticated language models appears to be a generallyuseful technique which can both improve upon direct estimation ¢ of -grams, and allows available higherlevellinguistic knowledge to be effectively integrated into speech decoding or other tasks that place strongconstraints on usable language models.7.8 Appendix: Related ProblemsWe have seen ¢ how -gram expectations for SCFGs can be obtained by solving linear systemsbased on the matrix L I A, where I is the identity matrix and A is the first-moment (expectation) matrix ofnonterminal occurrences for a single nonterminal expansion. As it turns out, a number of apparently unrelatedproblems arising in connection with SCFGs and other probabilistic grammars have solutions based on thissame matrix. These are briefly surveyed below, without detailed proofs.7.8.1 Expected string lengthTo compute the expected number of terminals in a string, the IL system A is solved for the right-handside vector containing the average number of terminals generated in a single production, for each nonterminal.For example, ifThe solution vector contains the expected lengths for the sublanguages generated by each of the2 N+9,) ¸ 0Í¾ 1.>!0Œ¾nonterminals. Thus, the expected sentence string length is the9-entry in the solution vector.The problems and its solution are easily generalized to obtain the expected number of terminals ofa particular type occuring in a string (Booth & Thompson 1973).7.8.2 Derivation entropyThe derivation entropy is the average number of bits required to specify a derivation from a SCFG.Is is computed from a right-hand side vector that contains the average negative log probabilities for theproductions of each LHS nonterminal. For example, based on
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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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CHAPTER 2. FOUNDATIONS 20Global mod
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CHAPTER 3. HIDDEN MARKOV MODELS 48c
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CHAPTER 3. HIDDEN MARKOV MODELS 50d
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correlation between initial and fin
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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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CHAPTER 4. STOCHASTIC CONTEXT-FREE
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104Chapter 5Probabilistic Attribute
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,1makingandCHAPTER 5. PROBABILISTIC
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CHAPTER 5. PROBABILISTIC ATTRIBUTE
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122Chapter 6Efficient parsing with
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1z1CHAPTER 6. EFFICIENT PARSING WIT
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and each state in set ( -ÏH ):6)
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Page 180 and 181: 168Chapter 7-grams from Stochastic
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Page 186 and 187: -grams CCHAPTER 7. -GRAMS FROM ST
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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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