Unified Bayesian theory of sparse linear regression with nuisance parameters

We study frequentist asymptotic properties of Bayesian procedures for high-dimensional Gaussian sparse regression when unknown nuisance parameters are involved. Nuisance can be finite-, high-, or infinite-dimensional. A mixture point masses at zero and continuous distributions is used the prior distribution on coefficients, appropriate parameters. The optimal posterior contraction hampered by presence parameters, also examined discussed. It shown that procedure yields strong model selection consistency. Bernstein-von Mises-type theorem coefficients obtained uncertainty quantification through credible sets with guaranteed coverage. Asymptotic numerous examples investigated using theory developed in this study.