A framework for estimating risk

Kroon, Rodney Stephen (2008-03)

Thesis (PhD (Statistics and Actuarial Sciences))--Stellenbosch University, 2008.


We consider the problem of model assessment by risk estimation. Various approaches to risk estimation are considered in a uni ed framework. This a discussion of various complexity dimensions and approaches to obtaining bounds on covering numbers is also presented. The second type of training sample interval estimator discussed in the thesis is Rademacher bounds. These bounds use advanced concentration inequalities, so a chapter discussing such inequalities is provided. Our discussion of Rademacher bounds leads to the presentation of an alternative, slightly stronger, form of the core result used for deriving local Rademacher bounds, by avoiding a few unnecessary relaxations. Next, we turn to a discussion of PAC-Bayesian bounds. Using an approach developed by Olivier Catoni, we develop new PAC-Bayesian bounds based on results underlying Hoe ding's inequality. By utilizing Catoni's concept of \exchangeable priors", these results allowed the extension of a covering number-based result to averaging classi ers, as well as its corresponding algorithm- and data-dependent result. The last contribution of the thesis is the development of a more exible shell decomposition bound: by using Hoe ding's tail inequality rather than Hoe ding's relative entropy inequality, we extended the bound to general loss functions, allowed the use of an arbitrary number of bins, and introduced between-bin and within-bin \priors". Finally, to illustrate the calculation of these bounds, we applied some of them to the UCI spam classi cation problem, using decision trees and boosted stumps. framework is an extension of a decision-theoretic framework proposed by David Haussler. Point and interval estimation based on test samples and training samples is discussed, with interval estimators being classi ed based on the measure of deviation they attempt to bound. The main contribution of this thesis is in the realm of training sample interval estimators, particularly covering number-based and PAC-Bayesian interval estimators. The thesis discusses a number of approaches to obtaining such estimators. The rst type of training sample interval estimator to receive attention is estimators based on classical covering number arguments. A number of these estimators were generalized in various directions. Typical generalizations included: extension of results from misclassi cation loss to other loss functions; extending results to allow arbitrary ghost sample size; extending results to allow arbitrary scale in the relevant covering numbers; and extending results to allow arbitrary choice of in the use of symmetrization lemmas. These extensions were applied to covering number-based estimators for various measures of deviation, as well as for the special cases of misclassi - cation loss estimators, realizable case estimators, and margin bounds. Extended results were also provided for strati cation by (algorithm- and datadependent) complexity of the decision class. In order to facilitate application of these covering number-based bounds,

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