  We propose a general framework using spike-and-slab prior distributions to
aid with the development of high-dimensional Bayesian inference. Our framework
allows inference with a general quasi-likelihood function. We show that highly
efficient and scalable Markov Chain Monte Carlo (MCMC) algorithms can be easily
constructed to sample from the resulting quasi-posterior distributions.
  We study the large scale behavior of the resulting quasi-posterior
distributions as the dimension of the parameter space grows, and we establish
several convergence results. In large-scale applications where computational
speed is important, variational approximation methods are often used to
approximate posterior distributions. We show that the contraction behaviors of
the quasi-posterior distributions can be exploited to provide theoretical
guarantees for their variational approximations. We illustrate the theory with
some simulation results from Gaussian graphical models, and sparse principal
component analysis.
