  Bounding chains are a technique that offers three benefits to Markov chain
practitioners: a theoretical bound on the mixing time of the chain under
restricted conditions, experimental bounds on the mixing time of the chain that
are provably accurate and construction of perfect sampling algorithms when used
in conjunction with protocols such as coupling from the past.
  Perfect sampling algorithms generate variates exactly from the target
distribution without the need to know the mixing time of a Markov chain at all.
We present here the basic theory and use of bounding chains for several chains
from the literature, analyzing the running time when possible. We present
bounding chains for the transposition chain on permutations, the hard core gas
model, proper colorings of a graph, the antiferromagnetic Potts model and sink
free orientations of a graph.
