This thesis responds to the challenges of using a large number, such as thousands, of features in regression and classification problems.
There are two situations where such high dimensional features arise. One is when high dimensional measurements are available, for example, gene expression data produced by microarray techniques. For computational or other reasons, people may select only a small subset of features when modelling such data, by looking at how relevant the features are to predicting the response, based on some measure such as correlation with the response in the training data. Although it is used very commonly, this procedure will make the response appear more predictable than it actually is. In Chapter $2$, we propose a Bayesian method to avoid this selection bias, with application to naive Bayes models and mixture models.
High dimensional features also arise when we consider high-order interactions. The number of parameters will increase exponentially with the order considered. In Chapter $3$, we propose a method for compressing a group of parameters into a single one, by exploiting the fact that many predictor variables derived from high-order interactions have the same values for all the training cases. The number of compressed parameters may have converged before considering the highest possible order. We apply this compression method to logistic sequence prediction models and logistic classification models.
We use both simulated data and real data to test our methods in both chapters.
[Software for: Chapter 2.1, Chapter 2.2, Chapter 3].