  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 theories developed in this study.
