The Joker: A Custom Monte Carlo Sampler for Binary-star and Exoplanet Radial Velocity Data

Given sparse or low-quality radial velocity measurements of a star, there are often many qualitatively different stellar or exoplanet companion orbit models that are consistent with the data. The consequent multimodality of the likelihood function leads to extremely challenging search, optimization, and Markov chain Monte Carlo (MCMC) posterior sampling over the orbital parameters. Here we create a custom Monte Carlo sampler for sparse or noisy radial velocity measurements of two-body systems that can produce posterior samples for orbital parameters even when the likelihood function is poorly behaved. The six standard orbital parameters for a binary system can be split into four nonlinear parameters (period, eccentricity, argument of pericenter, phase) and two linear parameters (velocity amplitude, barycenter velocity). We capitalize on this by building a sampling method in which we densely sample the prior probability density function (pdf) in the nonlinear parameters and perform rejection sampling using a likelihood function marginalized over the linear parameters. With sparse or uninformative data, the sampling obtained by this rejection sampling is generally multimodal and dense. With informative data, the sampling becomes effectively unimodal but too sparse: in these cases we follow the rejection sampling with standard MCMC. The method produces correct samplings in orbital parameters for data that include as few as three epochs. The Joker can therefore be used to produce proper samplings of multimodal pdfs, which are still informative and can be used in hierarchical (population) modeling. We give some examples that show how the posterior pdf depends sensitively on the number and time coverage of the observations and their uncertainties.