  This paper considers how to obtain MCMC quantitative convergence bounds which
can be translated into tight complexity bounds in high-dimensional {settings}.
We propose a modified drift-and-minorization approach, which establishes
generalized drift conditions defined in subsets of the state space. The subsets
are called the "large sets", and are chosen to rule out some "bad" states which
have poor drift property when the dimension of the state space gets large.
Using the "large sets" together with a "fitted family of drift functions", a
quantitative bound can be obtained which can be translated into a tight
complexity bound. As a demonstration, we analyze several Gibbs samplers and
obtain complexity upper bounds for the mixing time. In particular, for one
example of Gibbs sampler which is related to the James--Stein estimator, we
show that the number of iterations required for the Gibbs sampler to converge
is constant under certain conditions on the observed data and the initial
state. It is our hope that this modified drift-and-minorization approach can be
employed in many other specific examples to obtain complexity bounds for
high-dimensional Markov chains.
