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

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6tt,u ·¥¸¹u ,10ºtu ,2 10Yt ¸¹uu ,» 0Yt 1u ,» 0Yt 1u ,1 20Yt ¸¹uu ,1 20Yt ¸¹uu ,» 0 2u ,» 0ºt, u2 2 0 ¸¼u½CHAPTER 2. FOUNDATIONS 16S --> a 0.2--> b 0.2--> 0.1--> a S a 0.4--> b S b 0.1.Figure 2.2: Simple stochastic context-free grammar generating palindromesthe general sense introduced earlier. +9,21 . 450 is the sum of all the joint probabilities of random walks throughthe model (i.e., derivations) generating .In the example HMM, the string 1 is generated by only a single path through the model, andhence+-,µ. 45060£ 50 46 0£ 1375£The conditional probabilities in the product are the individual transition and emission probabilities. The factthat each is conditional only on the current state reflects the Markovian character of the generation process.2.3.5 Stochastic Context-free GrammarsStochastic context-free grammars (SCFGs) are the natural extension of context-free grammars(CFGs) to the probabilistic realm. We again present a simple example and leave the formal presentation toChapter 4.Each context-free productionis annotated with the conditional probabilitythat it is chosen among allpossible productions when the left-hand side nonterminal is expanded. Figure 2.2 shows a SCFG generatingall the palindromes over the E alphabet . A derivation in this case is the usual tree-structure (parse tree)arising from the nonterminal expansions, and its probability is the product of the probabilities of the rulesinvolves. Hence, the string abbba is generated with probability 0£ 4 ¾ 0£ 1 ¾ 0£ 1 ¾ 0£ 2, corresponding to theonly derivation of this string in the grammar.Notice how the notion of context-freeness has been extended to include probabilistic conditionalindependence of the expansion of a nonterminal from its surrounding context. This turns out to be crucial tokeep the computational properties of SCFGs reasonable, but it is also the major drawback of SCFGs whenusing them to model, say, natural language. We will return to this problem at various points in the followingchapters.
CHAPTER 2. FOUNDATIONS 172.4 Levels of Learning and Model Merging2.4.1 Beyond parameter estimationLearning of probabilistic language models from data involves the choice of a model that bestrepresents the distribution that generated the samples. We can identify three levels at which choices areinvolved:a) Choice of a model (¢ class -gram, finite-state, context-free, mixture of several of these, etc.).b) Choice of a model structure (how many states, nonterminals, what productions, etc.)c) Choice of (¢ parameters -gram probabilities, transition probabilities, etc.)So far we have briefly addressed (c), with maximum likelihood and EM estimators. These are infact the established methods for all of the language models discussed in this thesis. The main contribution ofthe thesis will be to (b), the choice of model structure. Level (a) is currently left to the human designer of alearning system. The fundamental problem of model design will be briefly discussed in in Chapter 8.The boundaries between (a) and (b), as well as between (b) and (c), are inherently fuzzy, dueto subsumption between representations. For example, in a finite-state model the presence or absenceof transitions can be represented either as part of the model structure (level b) or a continuous transitionprobability parameter (level c), where probability zero effectively represents the absence of a transition.These two representation correspond to very different learning approaches: to learn (b) the learner has tomake discrete decisions, whereas for (c) learning typically is accomplished by gradual adjustment of thecontinuous parameters (using EM, gradient ascent, etc.).Similarly, when faced with a choice between different model classes (level a) it is often possible todevise a generalized class that subsumes the alternatives in question. For example, the class of probabilisticcontext-free models contains finite-state models as a special case. One can then hope that whatever structurelearning method is used (level b) will pick out the right type of model. In practice this is not quite sostraightforward since one might end up with a model structure that doesn’t fall neatly into any of the targetclasses.2.4.2 Model mergingThe approach to the structural learning problem for probabilistic grammars pursued in this thesisis conceptually based on a rather general method recently advocated by Omohundro (1992) called modelmerging. Many of the key ingredients of this method are very old and can be found in algorithms from severaldisparate areas dealing with various incarnations of the data modeling task. Omohundro argues that thereare domain-independent principles at work that lend the model merging approach generality, efficiency and acertain degree of cognitive plausibility.The basic idea is that a model of a domain is constructed from submodels. When only a small amountof data is available, the submodels consist of the data points themselves with similarity-based generalization.
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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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