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

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CHAPTER 2. FOUNDATIONS 18As more data becomes available, submodels from more complex families may be constructed. The elementaryinduction step is to successively merge pairs of simple submodels to form more complex submodels. Becausethe new submodel is based on the combined data from the merged submodels, it may be reliably chosen froma more complex model space. This approach allows arbitrarily complex models to be constructed withoutoverfitting. The resulting models adapt their representational power to the structure of the training data.The search for submodels to merge is guided by an attempt to sacrifice as little of the samplelikelihood as possible as a result of the merging process. This search can be done very efficiently if (a) agreedy search strategy can be used, and (b) likelihood computations can be done locally for each submodeland don’t require global recomputation on each model update.2.4.3 A curve-fitting exampleThis idea can be illustrated with task from geometrical data modeling. Consider the problemof modeling a curve in the plane by a combination of straight line segments. The likelihood in this casecorresponds to the mean square error from each curve point to the nearest segment point. 8 A merging step inthis case consists of replacing two segments by a single segment. We always choose that pair such that themerged segment increases the error the least. Figure 2.3 shows the approximations generated by this strategy.It does an excellent job at identifying the essentially linear portions of the curve and puts the boundariesbetween component models at the corners. While not shown in the figure, as repeated merges take place,more data is available for each segment. This would allow us to reliably fit submodels more complex thanlinear segments, such as Bezier curves. It is possible to reliably induce a representation which uses linearsegments in some portions and higher order curves in others. Such models potentially have many parametersand would be subject to overfitting if they were learned directly rather than by going through merging steps.Model merging has an obvious converse in iterative model splitting. In the curve example, thistop-down approach would start with a single segment and repeatedly split it. This approach sometimes has tomake decisions too early and often misses the corners in the curve. Although this is clearly domain-dependent,our experience has been that modeling approaches based on splitting tend to fit the structure of a domain lesswell than those based on merging. Model splitting approaches for grammatical models have been proposedby various authors and will be discussed in Chapter 3.2.4.4 Knowing when to stopA crucial factor in model merging is the stopping criterion. Since each merging step is a potentialgeneralization step in the model space, the stopping criterion effectively determines how much generalizationfrom the observed data takes place.The criteria used can be of a theoretical, heuristic or practical nature. The following represents anon-exhaustive list of candidates:8 More precisely, the mean squared error is proportional to the log probability of the data under the assumption that each curvegenerates points normally distributed around them.
CHAPTER 2. FOUNDATIONS 19Error = 1 Error = 2 Error = 5 Error = 10 Error = 20Figure 2.3: Approximation of a curve by best-first merging of segment models.The top row shows the endpoints chosen by the algorithm at various levels of allowed error. Thebottom row shows the corresponding approximation to the curve.
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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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