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-grams CCHAPTER 7. -GRAMS FROM STOCHASTIC CONTEXT-FREE GRAMMARS 174The computation of prefix probabilities for SCFGs is generally useful for applications and has beensolved with the LRI algorithm (Jelinek & Lafferty 1991). In Chapter 6 we have seen how this computation canbe carried out efficiently for sparsely parameterized SCFGs using a probabilistic version of Earley’s parser.Computing suffix probabilities is obviously a symmetrical task; for example, one could create a ‘mirrored’SCFG (reversing the order of right-hand side symbols in all productions) and then run any prefix probabilitycomputation on that mirror grammar.Note that in the case of bigrams, only a particularly simple form of prefix/suffix probabilities arerequired, namely, the ‘left-corner’ ) ( 10 and ‘right-corner’ ( 20 probabilities, and , whichcan each be obtained from a single matrix inversion (Jelinek & Lafferty 1991), corresponding to the left-cornerX C +-,= +-,matrix © used in the probabilistic Earley parser (as well as the corresponding right-corner matrix).Finally, it is interesting to compare the relative ease with which one can solve the substringexpectation problem to the seemingly similar problem of finding substring probabilities: the probability that) generates (one or more instances of) ( . The latter problem is studied by Corazza et al. (1991), and shownto lead to a non-linear system of equations. The crucial difference here is that expectations are additive withrespect to the cases in Figure 7.1, whereas the corresponding probabilities are not, since the three cases canoccur in the same string.7.3.5 Ëcontaining string boundariesA complete ¢ -gram grammar includes strings delimited by a special marker denoting beginningand end of a string. In Section 2.2.2 we had introduced the symbol ‘$’ for this purpose.To generate expectations for ¢ÜL 1-grams adjoining the string boundaries, the original SCFGgrammar is augmented by a new top-level production9Ù¸$9$1£ 0óòis the old start symbol, and9%Ù becomes the start symbol of the augmented grammar. The algorithmis then simply applied to the augmented grammar to give the desired ¢ -gram probabilities including the ‘$’where9marker.7.4 Efficiency and Complexity IssuesSummarizing from the previous section, we can compute any ¢ -gram probability by solving twolinear systems of equations of the form (7.3), one with ( being the ¢ -gram itself and one for the , ¢¦L 10 -gramprefix ( 1 £££;(=$?1. The latter computation can be shared among all ¢ -grams with the same prefix, so thatessentially one system needs to be solved for each ¢ -gram we are interested in. The good news here is thatthe work required is linear in the number ¢ of -grams, and correspondingly limited if one needs probabilitiesfor only a subset of the ¢ possible -grams. For example, one could compute these probabilities on demandand cache the results.
where I is the identity matrix, A 6,,,””” ,,6Ny,) ¸ =>0+-,= , ,=iÌ¥0MNö¸¸What is the expected frequency of unigram ? Using the abbreviation Œ6|,199 9W9u 6 1 Lgtó,”,ö=,CHAPTER 7. -GRAMS FROM STOCHASTIC CONTEXT-FREE GRAMMARS 175in the formLet us examine these systems of equations one more time. Each can be written in matrix notationI L A0 c 6 b (7.6)c represents the vector of unknowns, 0 is a coefficient matrix, b 6)g0 . All of these are indexed by nonterminals )gÌ.(ë.We get0 is the right-hand side vector, and,>!Ì¥070 (7.7)6 y ”'’‘ž+-,+-,) ¸ (!0”'Ç‘ž+-,1) ¸ =>!0=>?y Xd10 (7.8)1 £££;(%ds0whereö,and 0 otherwise. The expression L I A arises from bringing the variablesto the other side in equation (7.3) in order to collect the coefficients.(-.>0)g)=ë0¦6 1 if ) 6À=,(+-,> CTg(%d #1 £££2(X CWe can see that all dependencies on the particular bigram, ( , are in the right-hand side vector b,while the coefficient matrix I L A depends only on the grammar. This, together with the standard methodof LU decomposition (see, e.g., Press et al. (1988)) enables us to solve for each bigram in time ð2 0 ,(ë.=ë0 and rather than the ð 30 standard for a full ( system being the number of nonterminals/variables). The LUdecomposition itself is cubic, but is incurred only once. The full computation is therefore dominated bythe quadratic effort of solving the system for each ¢ -gram. Furthermore, the quadratic cost is a worst-casefigure that would be incurred only if the grammar contained every possible rule; empirically this computationis linear in the number of nonterminals, for grammars that are sparse, i.e., where each nonterminal makesreference only to a bounded number of other nonterminals (independent of the total grammar size).7.5 Consistency of SCFGsBlindly applying the ¢ -gram algorithm (and many others) to a SCFG with arbitrary probabilitiescan lead to surprising results. Consider the following simple grammar1 ò(7.9)trówe see that)/.9%0 and equation 7.5,òiN¬0Ø60
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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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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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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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CHAPTER 4. STOCHASTIC CONTEXT-FREE
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==Ì==ÌCHAPTER 4. STOCHASTIC CONTE
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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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Page 144 and 145: ) In particular, the string probabi
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Page 162 and 163: description. Again, we ignore this
Page 164 and 165: for all pairs of states d =¸+= Š
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Page 178 and 179: = ¸Let t,t) ¸ .V=i£,t,t,6666yyyy
Page 180 and 181: 168Chapter 7-grams from Stochastic
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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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