CAREER: Heavy-Tailed Priors for Robust Bayesian Inference in Ecology, Machine Learning, and Astronomy

Heavy-tailed probability distributions are routine across many scientific disciplines, from astronomy to ecology and finance and network modeling. Such distributions are often utilized in statistical modeling to incorporate non-linearity, robustness to large observations, and sparsity in high-dimensional data. The overarching goal of this project is to build new scalable statistical methods that incorporate heavy-tailed prior distributions in three disparate application areas: independent component estimation that recovers independent, latent sources from their observed mixtures, astronomical distance estimation from parallax measurements, and statistical modeling of compositional data. The research will result in powerful Bayesian tools with rigorous theoretical justification. This project will also narrow the critical gap between methodological advances in statistics and the tools used by the scientific community and promote increased usage and transparency of state-of-the-art Bayesian tools. The research findings will be incorporated into various educational activities to engage K-12 students. The project will provide research opportunities and training for graduate students and will enhance undergraduate and graduate curricula, accompanied by a monograph. This project develops Bayesian methodologies to address three significant statistical challenges: (1) unifying feature extraction techniques via novel latent space representations in independent component analysis, (2) improving astronomical distance estimation by incorporating measurement errors and non-linear relationships in parallax data, and (3) constructing prior distributions tailored for high-dimensional simplex-valued data that can adapt to arbitrary sparsity and dependence patterns. By leveraging heavy-tailed priors within hierarchical models, this work provides a new framework for controlling higher-order moments in blind source separation and as mixing densities for normal scale mixtures for handling non-linearity, robustness, and sparsity. The methods to be developed will be rigorously tested in applications spanning astronomy, blind source separation, community detection, and ecological modeling of species diversity and affinity, demonstrating their broad utility. The results will be disseminated through peer-reviewed publications in statistics, machine learning, and other scientific journals, and software implementations will be openly accessible as R packages that benefit the wider quantitative science community. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.