Leveraging Structural Information in Regression Tree Ensembles

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A common task in statistics is prediction; for example, a practitioner may be interested in predicting the presence of a disease given genetic information about an individual. Due to recent advances in data collection, frequently one has access to datasets which contain a massive number of predictors, but with correspondingly few subjects. This setting is generally referred to as the "big P, small n" scenario. Drawing meaningful conclusions under such circumstances is generally impossible unless the underlying data satisfy certain structural assumptions. The simplest such structural assumption is that only a small number of the predictors are relevant; in this setting, finding the useful predictors corresponds to finding a so-called "needle in a haystack." The goal of this project is to construct procedures which adapt to this, and other, structural assumptions. The project will focus on methods based on decision trees, which are flowchart-like structures in which predictions are based on whether the predictors satisfy various rules. Usually an ensemble of decision trees are constructed, with the predictions for each individual tree averaged. While decision tree ensembles are frequently used with high dimensional data, it is unclear to what extent they adapt to the structural properties of the data. This project will show that, in practice, off-the-shelf decision tree ensembling methods do not adapt to common structural assumptions, and will develop new methods which do. In addition to developing methods with strong theoretical support, this project will support the development of an R package to give practitioners easy access to our methodology. The PI will develop Bayesian methods for incorporating structural information into tree-based ensemble methods, and establish theoretically the benefit of making use of this additional information. This forms a nonparametric counterpart to the parametric approaches used in linear models, such as the lasso, graphical lasso, or group lasso; Bayesian approaches in the parametric setting include the use of variable selection priors, such as spike-and-slab priors and global-local shrinkage priors. Structural information will be incorporated by modifying the commonly used priors on decision tree ensembles so that the prior is concentrated on models which satisfy the desired structure. The PI will first investigate the theoretical properties of a sparsity inducing prior which is designed to eliminate unnecessary predictors. Sparsity here is obtained by applying a sparsity inducing Dirichlet prior to the a priori probability that a given branch is associated to a given predictor. This prior will be extended to allow for grouped variable selection in a similar manner to the group lassoby considering the class of Dirichlet tree priors, and further to accommodate graphical structures in the predictors through sparsity inducing logistic normal priors. Additionally, the PI will develop computationally efficient Markov chain Monte Carlo algorithms to fit the resulting models. Compared to existing methods, these structural priors will be shown to lead to substantial gains in predictive accuracy, and to more accurate scientific discovery.