The discovery of multiple planets orbiting the Pulsar PSR B1257+12 (Wolszczan & Frail, 1992), ushered in an exciting new era of astronomy. By 2017, over 3500 exoplanets had been discovered by a great variety of techniques, including precision radial velocity. The increase in parameters needed to model multiple planetary systems is motivating efforts to improve the statistical tools for analyzing the radial velocity data. Much of the recent work has highlighted a Bayesian MCMC approach as a way to better understand parameter uncertainies and degeneracies.
I developed (Gregory 2005, 2006, 2007, 2008, 2010, 2011, 2013) a Bayesian nonlinear model fitting program based on a new fusion MCMC algorithm. The algorithm incorporates parallel tempering, genetic crossover operations, and simulated annealing via a unique adaptive control system that automates the tuning of proposal distributions for efficient exploration of the parameter space even for highly correlated parameters. Each of these methods was designed to facilitate the detection of a global maximum in a fitness criteria. By combining all three the fusion MCMC greatly increases the probability of realizing this goal. When applied to the Kepler problem it acts as a powerful multi-planet Kepler periodogram which provides full Bayesian posterior parameter probability density distributions for all the orbital elements that can be determined from precision radial velocity data.
In 2015, I developed a new Apodized Keplerian (AK) model to distinguish planetary signals from stellar activity induced RV signals (Gregory 2016). Recent tests indicate that the AK method is able to achieve a reduction in stellar activity noise by a factor of approximately 6.
The Bayesian framework allows a comparison of models with different numbers of planets by comparing the marginal likelihood for each model. The development of alternative robust schemes for computing the marginal likelihoods is an active research topic (Ford & Gregory 2006). See my text book supplement for a discussion of a new but conceptually simple method called Nested Restricted Monte Carlo (NRMC), along with a detailed comparison to two other methods in an exoplanet application.