Probabilities of exoplanet signals from posterior samplings

¹ University of Hertfordshire, Centre for Astrophysics Research, Science and Technology Research Institute, College Lane, AL10 9AB, Hatfield, UK
e-mail: m.tuomi@herts.ac.uk; mikko.tuomi@utu.fi
² University of Turku, Tuorla Observatory, Department of Physics and Astronomy, Väisäläntie 20, 21500 Piikkiö, Finland

Received: 16 September 2011
Accepted: 23 June 2012

Abstract

Aims. Estimating the marginal likelihoods is an essential feature of model selection in the Bayesian context. It is especially crucial to have good estimates when assessing the number of planets orbiting stars and different models explain the noisy data with different numbers of Keplerian signals. We introduce a simple method for approximating the marginal likelihoods in practice when a statistically representative sample from the parameter posterior density is available.

Methods. We use our truncated posterior mixture estimate to receive accurate model probabilities for models with different numbers of Keplerian signals in radial velocity data. We test this estimate in simple scenarios to assess its accuracy and rate of convergence in practice when the corresponding estimates calculated using the deviance information criterion can be applied to obtain trustworthy model comparison results. As a test case, we determine the posterior probability of a planet orbiting HD 3651 given Lick and Keck radial velocity data.

Results. The posterior mixture estimate appears to be a simple and an accurate way of calculating marginal integrals from posterior samples. We show that it can be used in practice to estimate the marginal integrals reliably, given a suitable selection of the parameter λ, which controls the estimate’s accuracy and convergence rate. It is also more accurate than the one-block Metropolis-Hastings estimate, and can be used in any application because it is based on assumptions about neither the nature of the posterior density nor the amount of either data or parameters in the statistical model.