  This paper introduces a concept of approximate spectral gap to analyze the
mixing time of Markov Chain Monte Carlo (MCMC) algorithms for which the usual
spectral gap is degenerate or almost degenerate. We use the idea to analyze a
class of MCMC algorithms to sample from mixtures of densities. As an
application we study the mixing time of a Gibbs sampler for variable selection
in linear regression models. Under some regularity conditions on the signal and
the design matrix of the regression problem, we show that for well-chosen
initial distributions the mixing time of the Gibbs sampler is polynomial in the
dimension of the space.
