Stellar radial velocity (RV) measurements have proven to be a very successful method for detecting extrasolar planets. Analysing RV data to determine the parameters of the extrasolar planets is a significant statistical challenge owing to the presence of multiple planets and various degeneracies between orbital parameters. Determining the number of planets favoured by the observed data is an even more difficult task. Bayesian model selection provides a mathematically rigorous solution to this problem by calculating marginal posterior probabilities of models with different number of planets, but the use of this method in extrasolar planetary searches has been hampered by the computational cost of the evaluating Bayesian evidence. Nonetheless, Bayesian model selection has the potential to improve the interpretation of existing observational data and possibly detect yet undiscovered planets. We present a new and efficient Bayesian method for determining the number of extrasolar planets, as well as for inferring their orbital parameters, without having to calculate directly the Bayesian evidence for models containing a large number of planets. Instead, we work iteratively and at each iteration obtain a conservative lower limit on the odds ratio for the inclusion of an additional planet into the model. We apply this method to simulated data-sets containing one and two planets and successfully recover the correct number of planets and reliable constraints on the orbital parameters. We also apply our method to RV measurements of HD 37124, 47 Ursae Majoris and HD 10180. For HD 37124, we confirm that the current data strongly favour a three-planet system. We find strong evidence for the presence of a fourth planet in 47 Ursae Majoris, but its orbital period is suspiciously close to one year, casting doubt on its validity. For HD 10180 we find strong evidence for a six-planet system.
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