Bayesian inference and topological data analysis provide established machine learning tools. However, deep learning poses scalability challenges to these tools. My research focuses on Bayesian deep learning (BDL) and topological deep learning (TDL), which aim to make Bayesian and topological approaches to deep learning feasible. In the contemporary era of AI, BDL models can quantify uncertainties and assess risks, and TDL models can extract structure from big data. Healthcare is a safety-critical domain of application that interests me, as uncertainty quantification and elicitation of structure from biomedical data matter.