  We discuss a Bayesian approach to the analysis of radial velocities in planet
searches. We use a combination of exact and approximate analytic and numerical
techniques to efficiently evaluate chi-squared for multiple values of orbital
parameters, and to carry out the marginalization integrals for a single planet
including the possibility of a long term trend. The result is a robust
algorithm that is rapid enough for use in real time analysis that outputs
constraints on orbital parameters and false alarm probabilities for the planet
and long term trend. The constraints on parameters and odds ratio that we
derive compare well with previous calculations based on Markov Chain Monte
Carlo methods, and we compare our results with other techniques for estimating
false alarm probabilities and errors in derived orbital parameters. False alarm
probabilities from the Bayesian analysis are systematically higher than
frequentist false alarm probabilities, due to the different accounting of the
number of trials. We show that upper limits on the velocity amplitude derived
for circular orbits are a good estimate of the upper limit on the amplitude of
eccentric orbits for eccentricities less than about 0.5.
