Model Selection Methods for Linear Regression and Phylogenetic Reconstruction
Model selection is the task of selecting from a collection of alternative explanations (often probabilistic models) the one that is best suited for a given data set. This thesis studies model selection methods for two domains, linear regression and phylogenetic reconstruction, focusing particularly on situations where the amount of data available is either small or very large. In linear regression, the thesis concentrates on sequential methods for selecting a subset of the variables present in the data. A major result presented in the thesis is a proof that the Sequentially Normalized Least Squares (SNLS) method is consistent, that is, if the correct answer (i.e. the so called true model) exists, then the method will find it with probability that approaches one as the amount of data increases. The thesis also introduces a new sequential model selection method that is an intermediate form between SNLS and the Predictive Least Squares (PLS) method. In addition, the thesis shows how these methods may be used to enhance a novel algorithm for removing noise from images. For phylogenetic reconstruction, that is, the task of inferring ancestral relations from genetic data, the thesis concentrates on the Maximum Parsimony (MP) approach that tries to find the phylogeny (family tree) which minimizes the number of evolutionary changes required. The thesis provides values for various numerical indicators that can be used to assess how much