Bayesian Decision Theoretic Methods for Some High-Dimensional Multiple Inference Problems

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The project covers some outstanding and important inference problems that statisticians face when analyzing high-dimensional data from brain imaging, next generation sequencing, atmospheric science, astronomical studies, and many other scientific investigations. Miss-detecting a strong signal in these experiments, particularly when the signals are sparse, is often a more severe error than miss-detecting a weak signal, and this error gets more severe as the signal gets stronger. This is an important issue which has not been fully utilized in the existing procedures designed for simultaneous testing of multiple hypotheses. Also, selective inference using multiple confidence intervals is an emerging area of statistical research whose importance is being realized very recently. However, while analyzing high-dimensional data with sparse signals, the existing intervals designed to provide estimates of the selected significant signals can become non-informative in the sense of miss-covering the true signal or covering zero too often if the sparse nature of the data is not properly taken into account. This research project seeks to develop new and innovative methods taking a Bayesian decision theoretic viewpoint which is particularly well suited to tackle these issues. It focuses on the following two broad areas of research: (i) Developing new multiple testing methods controlling false discoveries incorporating the severity of type II errors, and (ii) developing new multiple confidence intervals for selected parameters under zero-inflated mixture prior. This project will be expected to have a broad impact on the theory and practice of statistics. It can produce novel methodologies to detect true signals in modern and high-dimensional scientific investigations, and pave the way for better use of statistics towards meeting modern societal and scientific needs. For instance, understanding vegetation changes under seasonal variability is crucial for more effective land use management when coping with climate changes and food security. This project can potentially offer new methodologies towards addressing that sustainability issue. Also, there is an increasing demand for sophisticated statistical tools to have better understanding of astronomical behaviors based on the influx of data created by the advent of new technologies. Again, this project can potentially meet that demand. The results will be disseminated through presentations and discussions at national and international conferences, and visits to other institutions. The software to be developed under this project will be made available, free of charge, to the scientific community.