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Modern science and engineering are producing data at unprecedented rates, and the data being generated is noisier and more sparse than in the past. In this talk, I will present a survey of my research describing applications of Bayesian inference and Topological Data Analysis (TDA) to noisy, large, sparse data and combinations thereof. First, we will discuss a Bayesian state-space model of the first six months of the Covid-19 pandemic, when data was sparse and unreliable, with a focus on model selection criteria. Next, I will present work coupling variational Bayesian inference and TDA, reconstructing the geometry of the atoms in a material and determining if any preferential atomic ordering exists from a large, sparse, and noisy materials dataset. We conclude by combining Bayesian and topological techniques to select relevant features from a deep learning model developed for biomedical information extraction. The latter two projects demonstrate the breadth of knowledge available from coupling these techniques and the ability to provide an holistic picture of the data to domain scientists.