  Convergence rate analyses of random walk Metropolis-Hastings Markov chains on
general state spaces have largely focused on establishing sufficient conditions
for geometric ergodicity or on analysis of mixing times. Geometric ergodicity
is a key sufficient condition for the Markov chain Central Limit Theorem and
allows rigorous approaches to assessing Monte Carlo error. The sufficient
conditions for geometric ergodicity of the random walk Metropolis-Hastings
Markov chain are refined and extended, which allows the analysis of previously
inaccessible settings such as Bayesian Poisson regression.
  The key technical innovation is the development of explicit drift and
minorization conditions for random walk Metropolis-Hastings, which allows
explicit upper and lower bounds on the geometric rate of convergence. Further,
lower bounds on the geometric rate of convergence are also developed using
spectral theory. The existing sufficient conditions for geometric ergodicity,
to date, have not provided explicit constraints on the rate of geometric rate
of convergence because the method used only implies the existence of drift and
minorization conditions.
  The theoretical results are applied to random walk Metropolis-Hastings
algorithms for a class of exponential families and generalized linear models
that address Bayesian Regression problems.
