  Precise radial velocity measurements have led to the discovery of ~170
extrasolar planetary systems. Understanding the uncertainties in the orbital
solutions will become increasingly important as the discovery space for
extrasolar planets shifts to planets with smaller masses and longer orbital
periods. The method of Markov chain Monte Carlo (MCMC) provides a rigorous
method for quantifying the uncertainties in orbital parameters in a Bayesian
framework (Ford 2005a). The main practical challenge for the general
application of MCMC is the need to construct Markov chains which quickly
converge. The rate of convergence is very sensitive to the choice of the
candidate transition probability distribution function (CTPDF). Here we explain
one simple method for generating alternative CTPDFs which can significantly
speed convergence by one to three orders of magnitude. We have numerically
tested dozens of CTPDFs with simulated radial velocity data sets to identify
those which perform well for different types of orbits and suggest a set of
CTPDFs for general application. Additionally, we introduce other refinements to
the MCMC algorithm for radial velocity planets, including an improved treatment
of the uncertainties in the radial velocity observations, an algorithm for
automatically choosing step sizes, an algorithm for automatically determining
reasonable stopping times, and the use of importance sampling for including the
dynamical evolution of multiple planet systems. Together, these improvements
make it practical to apply MCMC to multiple planet systems. We demonstrate the
improvements in efficiency by analyzing a variety of extrasolar planetary
systems.
