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

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Consider the following problem: starting from a ) nonterminal , how many nonterminals type= of”‘0 be the matrix formed by these expectations (assuming they exist). It is easy,are generated? Let N 6,?,?¢I L A0,n‘6 y‘I L A0 x 6 b £,?L A0 I,CHAPTER 7. -GRAMS FROM STOCHASTIC CONTEXT-FREE GRAMMARS 180the right-hand side entry ) for L+-,) ¸is log +-, ) ¸ .B0 L+-,) ¸ ˜M0 log +-, ) ¸ ˜M0 .The derivation entropy of the grammar is the solution vector entry indexed by the start symbol..B07.8.3 Expected number of nonterminal occurrencesto verify that N satisfies the familiar recurrence N 6 I N AN, from which we get N 61 .Another way to look at this outcome is that the expected number of nonterminals of type=,represented by the column vector in n‘N, is a problem of the same type as those described above:where is i‘following sense.the=-column of the identity matrix.In fact, the nonterminal expectations are the most general of all the problems of this type, in thei‘All the other quantities considered here, and conversely all those with a linear systemsolution defined by a right-hand side vector b and the coefficient matrix I L A, can be represented as weightedsums over the nonterminal expectations N, where the weighting is provided by the vector b.For example, the expected length 1 ”first principles) to be the weighted sumof the strings generated by )nonterminal can be seen (from61 ”¢¢ where is average number of=’s generated ) by , and the expected number of terminals produced(in a”‘‘single derivation step). Written in matrix form this becomes”‘ ‘by=x 6 Nbisand using N 6I L A01 we get the system used above (Section 7.8.1) for solving x,Thus, derivation entropy, ¢ -gram expectations, etc. all turn out to be linear weighted sums of thenonterminal expectations N. Needless to say, these quantities are well-defined if and only if the underlyingSCFG is consistent.7.8.4 Other grammar typesThe significance IL of A extends to all types of probabilistic grammars based on Markov or branchingprocesses over states. For example, in HMMs all of the above computations apply when ‘nonterminal’ isreplaced by ‘state’ where appropriate. The first-moment matrix A for HMMs is just the transition matrix, andN 6I L A01 expresses the expected number of visits to a state.
181Chapter 8Future directionsIn this final chapter we suggest a number of potential continuations of the work described in thisdissertation. Probabilistic language modeling has certainly proven to be a theoretically and practically usefulapproach, including the model merging paradigm for structural learning studied here. It has given rise toconcrete applications (in the case of HMMs) and gives new perspectives on some linguistic issues, as we havetried to show mainly in our discussion of the SCFG and PAG modeling frameworks.However, and not surprisingly, the work to date raises more questions than it provides definitiveanswers. Apart from model-specific problems discussed in the preceding chapters, a number of morefundamental issues can be identified. Invariably, these lead to worthwhile avenues for future work.8.1 Formal characterization of learning dynamicsIt would be extremely useful to be able to predict the sample complexity of a given target grammarand merging-based learning algorithm, i.e., the number of samples required to reliably learn the targetgrammar. As a first step, this would include a prediction of how many samples are needed to make the targetgrammar (or one equivalent to it) the one with globally optimal posterior probability. A further, more difficultproblem is to predict when a limited search procedure of specified type would be able to actually find thatglobal optimum.Within this general problem of sample complexity it is also interesting to draw a distinctionbetween sample structure and sample distribution. In other words, how sensitive is a learner to a sample thatis representative regarding the possible strings of the language, but exhibits different frequency statistics? 11 This question was pointed out by Robert Wilensky.
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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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correlation between initial and fin
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domain. 3 In short, we will leave o
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104Chapter 5Probabilistic Attribute
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CHAPTER 5. PROBABILISTIC ATTRIBUTE
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122Chapter 6Efficient parsing with
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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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