Bayesian regression on numerous tensors

This is a recent thesis on statistics:

Bayesian Modeling of Multi-Source Multi-Way Data

by  Jonathan Kim

University of Minnesota 

The reason why we bring this here is not only that the dedication quotes from Emily Brontë's The Philosopher. It's because BRONTe has now a new meaning: Bayesian regression on numerous tensors.

Biomedical research often involves data collected from multiple sources and these sources often have a multi-way (i.e.. multidimensional tensor) structure. Existing methods that can accommodate multi-source or multi-way data have various limitations on the exact structure of the data they are able to accommodate and in the type of predictions, if any, they are able to produce. Furthermore, few of these methods are able to handle data that are simultaneously multi-source and multi-way. We first introduce two such multi-source and multi-way datasets of molecular and hematological data from multiple sources, each measured over multiple developmental time points and in multiple tissues, as predictors of early-life iron deficiency (ID) in a rhesus monkey model. We describe preliminary analyses that were conducted on these datasets using existing methods. We then develop a Bayesian linear model that can perform prediction on a binary or continuous outcome and can accommodate data that are both multi-source and multiway. We use a linear model with a low-rank structure on the coefficients to capture multi-way dependence and model the variance of the coefficients separately across each source to infer their relative contributions. Conjugate priors facilitate an efficient Gibbs sampling algorithm for posterior inference, assuming a continuous outcome with normal errors or a binary outcome with a probit link. Simulations demonstrate that our mode performs as expected in terms of misclassification rates and correlation of estimated coefficients with true coefficients, with large gains in performance by incorporating multiway structure and modest gains when accounting for differing signal sizes across the different sources. Moreover, it provides robust classification of ID monkeys for one of our motivating datasets. Finally, we propose a flexible method called Bayesian regression on numerous tensors (BRONTe) that can predict a continuous or binary outcome from data that are collected from an arbitrary number of sources with multi-way tensor structures of arbitrary, not necessarily equal, orders. Additionally, BRONTe is able to accommodate data where some sources partially share features within a dimension. Simulations show BRONTe to perform well at prediction when the data sources are of unequal dimensions. In an application to our other motivating dataset on multi-way measures of metabolomics and hematology parameters, BRONTe was capable of robust classification of early-life iron deficiency.