Advances in Bayesian inference for graphical models and beyond Scalable algorithms and their applications

Data often provide incomplete or ambiguous information. It is therefore important, not only to estimate relationships between variables, but also to quantify uncertainty about these estimates. This thesis focuses on Bayesian uncertainty quantification. The Bayesian approach offers a wide variety of uncertainty quantification, holds for small sample sizes, and is able to incorporate prior knowledge. Despite these advantages, Bayesian approaches are often considered computationally expensive and difficult to apply.
This thesis challenges that perception by reviewing existing methods, improving their accessibility, and developing new algorithms that are accurate, scalable, and computationally efficient.
We start with undirected Gaussian graphical models (GGMs). We provide a review of existing Bayesian methods in this field and present two new algorithms. These methods are able to quantify uncertainty on instances with a thousand variables in under an hour.
We then move to Gaussian copula graphical models (GCGMs) and apply them to the field of Alzheimer's disease (AD). This leads to new insights into AD pathogenesis. Lastly, we design a general algorithm for Bayesian inference on all models that operate on a binary model space, including all graphical models (e.g. GGMs, GCGMs, Ising models), but also variable selection. Our algorithm does not depend on local moves or accept/reject steps but instead relies on a diminishing step length to ensure convergence to the model posterior. This leads to a massive reduction in computational cost without sacrificing accuracy.