  Radial velocity (RV) planet searches are increasingly finding planets with
small velocity amplitudes, with long orbital periods, or in multiple planet
systems. Bayesian inference has the potential to improve the interpretation of
existing observations, the planning of future observations and ultimately
inferences concerning the overall population of planets. In recent years, the
refinement of Markov chain Monte Carlo (MCMC) algorithms has made it practical
to accurately characterize orbital parameters and their uncertainties from RV
observations of single-planet and weakly interacting multiple-planet systems.
Unfortunately, MCMC is not sufficient for Bayesian model selection, i.e.,
comparing the marginal posterior probability of models, as is necessary to
determine how strongly the observational data favor a model with n+1 planets
over a model with just n planets. Many of the obvious estimators for the
marginal posterior probability suffer from poor convergence properties. We
compare several estimators of the marginal likelihood and feature those that
display desirable convergence properties based on the analysis of a sample data
set for HD 88133b. We find that methods based on importance sampling are most
efficient, provided that a good analytic approximation of the posterior
probability distribution is available. We present a simple algorithm for using
a sample from the posterior to construct a mixture distribution that
approximates the posterior and can be used for importance sampling and Bayesian
model selection. We conclude with some suggestions for the development and
refinement of computationally efficient and robust estimators of marginal
posterior probabilities.
