We study frequentist asymptotic properties of Bayesian procedures for high-dimensional Gaussian sparse regression when unknown nuisance parameters are involved. Nuisance parameters can be finite-, high-, or infinite-dimensional. A mixture of point masses at zero and continuous distributions is used for the prior distribution on sparse regression coefficients, and appropriate prior distributions are used for nuisance parameters. The optimal posterior contraction of sparse regression coefficients, hampered by the presence of nuisance parameters, is also examined and discussed. It is shown that the procedure yields strong model selection consistency. A Bernstein-von Mises-type theorem for sparse regression coefficients is also obtained for uncertainty quantification through credible sets with guaranteed frequentist coverage. Asymptotic properties of numerous examples are investigated using the theory developed in this study.
Research is partially supported by a Faculty Research and Professional Development Grant from College of Sciences of North Carolina State University and by ARO grant number 76643-MA.
"Unified Bayesian theory of sparse linear regression with nuisance parameters." Electron. J. Statist. 15 (1) 3040 - 3111, 2021. https://doi.org/10.1214/21-EJS1855