Bayesian Statistics using Stellar Radial Velocity Data

One of the most important discoveries of the last two decades in astrophysics was that of extrasolar planets, or exoplanets. The discovery of exoplanets not only has expanded enormously our knowledge about planetary systems, but also has driven the advancement of computational and numerical techniques. In this thesis, I present improvements and applications of Markov Chain Monte Carlo methods for the study of exoplanets in the framework of Bayesian inference and model selection using stellar radial velocity data. For Bayesian inference, an ensemble sampler respecting affine invariance is introduced to extract orbital parameters from radial velocity data. This sampler has only one tuning parameter, hence is very easy to automate. The autocorrelation time of this sampler is approximately the same for all the model parameters and in many cases far smaller than Metropolis-Hastings. A clustering technique based on the likelihood of the walkers in the ensemble is integrated to deal approximately with local minima. For model selection, a geometric-path Monte Carlo method, inspired by multi-canonical Monte Carlo, is applied to the evaluation of the fully marginalized likelihood, or Bayesian evidence, which is of central importance in Bayesian model selection but extremely challenging to compute. This algorithm is quite fast and easy to implement, and it produces a justified uncertainty estimate on the fully marginalized likelihood. I have successfully evaluated the fully marginalized likelihood of multi-companion models fitting for radial velocity data using this algorithm. A more sophisticated noise model using a Gaussian process with a non-trivial covariance structure to model stochastic stellar oscillations is also presented. It is shown that the orbital parameter inferences in real data are improved with the use of this noise model.