Directed Graphs and Applications

Abstract

Directed graphs have widespread applicability, so there is continued need for both computational and theoretical improvements. We extend the standard methodology in three different ways, which constitute the main chapters of the thesis.

In spatial settings, information about data collection location is often ignored. We apply nearest neighbor ideas to an underlying directed graph of the data for covariance estimation. By making the underlying graph sparse, estimation is scalable and more stable for a small number of observations. Novel priors to encourage stability are developed in the Bayesian paradigm. Our methods are highly scalable, flexible, and parallelizable. We conduct numerical comparisons and apply our methodology to climate-model output, enabling statistical emulation of an expensive physical model.

Next, in many healthcare settings, maintaining data privacy becomes paramount. Without sharing the data itself, local models could be vastly different; by pooling even model information, the overall model should be more unified and robust. This novel area of interest is called federated learning. We establish sparse regression and directed graphical modelling techniques for this task. Results show that our model is similar to modelling all of the data jointly but is faster and maintains the local privacy. We illustrate our method on both hospital COVID data and cancer genetics data.

Finally, we relax the assumption of no cycles as many realistic systems have feedback loops. Although it is less computationally scalable when allowing cycles, we exploit various mathematical formulas to help improve computation. By allowing cycles, we are also able to use simpler priors and not require exact zeros as are necessary in acyclic modelling. We design a Bayesian method for estimating nonlinear directed graphs while allowing cycles under the assumption that the system is in equilibrium. We demonstrate the nonlinear modelling capacity with B-splines, but the only requirement is to have the gradient available in closed form. Overall, we advanced directed graph methodology in three ways with Bayesian methodology.