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 ^ { 2 } 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 e \la 0.5 .