Improving the Efficiency of Markov Chain Monte Carlo for Analyzing the Orbits of Extrasolar Planets

By Eric B. Ford. 2005

362 citations recorded by the source corpus.

Precise radial velocity measurements have led to the discovery of ~170 extrasolar planetary systems. Understanding the uncertainties in the orbital solutions will become increasingly important as the discovery space for extrasolar planets shifts to planets with smaller masses and longer orbital periods. The method of Markov chain Monte Carlo (MCMC) provides a rigorous method for quantifying the uncertainties in orbital parameters in a Bayesian framework (Paper I). The main practical challenge for the general application of MCMC is the need to construct Markov chains that quickly converge. The rate of convergence is very sensitive to the choice of the candidate transition probability distribution function (CTPDF). Here we explain one simple method for generating alternative CTPDFs that can significantly speed convergence by 1-3 orders of magnitude. We have numerically tested dozens of CTPDFs with simulated radial velocity data sets to identify those that perform well for different types of orbits and suggest a set of CTPDFs for general application. In addition, we introduce other refinements to the MCMC algorithm for radial velocity planets, including an improved treatment of the uncertainties in the radial velocity observations, an algorithm for automatically choosing step sizes, an algorithm for automatically determining reasonable stopping times, and the use of importance sampling for including the dynamical evolution of multiple-planet systems. Together, these improvements make it practical to apply MCMC to multiple-planet systems. We demonstrate the improvements in efficiency by analyzing a variety of extrasolar planetary systems.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/1347c31423bbb901c2f5acd3720d85998c48ab6b

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Citations in the corpus

The LCES HIRES/Keck Precision Radial Velocity Exoplanet Survey (2017, methodology): “…chain members (considered to be the burn-in phase in which the chain identified the most probable regions in the parameter space), we divided the remaining chain into three parts and tested non-convergence by applying the Gelman–Rubin statistics for all parameters (Gelman et al. 2003; Ford 2006).”
RadVel: The Radial Velocity Modeling Toolkit (2018, background): “The chains are deemed well-mixed and the MCMC run is halted when the G-R statistic is less than 1.01 and the number of independent samples (Tz statistic, Ford 2006) is greater than 1000 for all free parameters for at least 5 consecutive checks.”
Efficient identification of exoplanetary transit candidates from SuperWASP light curves (2007, methodology): “Ford (2006) and Gregory (2007) have applied MCMC to the derivation of orbital parameters and detection of additional planets from radialvelocity curves.”
The GAPS Programme with HARPS-N at TNG - XIV. Investigating giant planet migration history via improved eccentricity and mass determination for 231 transiting planets (2017, positive): “The uniform prior on the eccentricity was ensured by weighting the stepping probability by the Jacobian of the transformation from [ e cos ω, e sin ω ] to [ e , ω ] (Ford 2006).”
FRIENDS OF HOT JUPITERS. I. A RADIAL VELOCITY SEARCH FOR MASSIVE, LONG-PERIOD COMPANIONS TO CLOSE-IN GAS GIANT PLANETS (2013, methodology): “We considered the chains wellmixed and halted the DE-MCMC run when the number of independent draws (Tz, as defined in Ford 2006) was greater than 1000 and the Gelman-Rubin statistic (Gelman et al. 2003; Ford 2006; Holman et al. 2006) was within 1% of unity for all parameters.”
THE KEPLER-10 PLANETARY SYSTEM REVISITED BY HARPS-N: A HOT ROCKY WORLD AND A SOLID NEPTUNE-MASS PLANET (2014, neutral): “This modification allows a more efficient exploration of the parameter space in the case of small eccentricities (Ford 2006), which is the case for Kepler-10b and c (Fogtmann-Schulz et al. 2014).”
Origins of Eccentric Extrasolar Planets: Testing the Planet-Planet Scattering Model (2007, methodology): “We assume the published model type (i.e., the number of planets and whether there is a long term trend) and apply the Markov chain Monte Carlo algorithm described in Ford (2005, 2006).”
THE HAWAII INFRARED PARALLAX PROGRAM. I. ULTRACOOL BINARIES AND THE L/T TRANSITION, (2012, methodology): “This initial chain was run according to the procedure outlined by Ford (2006) in which each value of β is periodically adjusted until the acceptance rate for that parameter comes within some tolerance (we chose 5”
Planets and Stellar Activity: Hide and Seek in the CoRoT-7 system (2013, methodology): “Because it is difficult to constrain the argument of periastron for planets in loweccentricity orbits, we introduce two parameters Ck and Sk (Ford 2006).”
RETIRED A STARS AND THEIR COMPANIONS. III. COMPARING THE MASS–PERIOD DISTRIBUTIONS OF PLANETS AROUND A-TYPE STARS AND SUN-LIKE STARS (2009, methodology): “We use a Markov-Chain Monte Carlo (MCMC) algorithm following the description in Ford (2005) to estimate our orbital parameter uncertainties (see also Ford 2006, Ford & Gregory 2007).”
Analytic Approximations for Transit Light-Curve Observables, Uncertainties, and Covariances (2008, methodology): “(29), |J|, is also useful in translating a priori probability distributions from one parameter set to the other [see Burke et al. (2007) or Ford (2006) for an example of how this is done in practice].”
TAU-REX I: A NEXT GENERATION RETRIEVAL CODE FOR EXOPLANETARY ATMOSPHERES (2014, neutral): “MCMC routines are commonly used in the field of extrasolar planets (e.g. Ford 2006; Burke et al. 2010; Bakos et al. 2007; Knutson et al. 2007; Cameron et al. 2007; Charbonneau et al. 2009; Bean et al. 2010; Kipping & Bakos 2011; Gregory 2011; Crouzet et al. 2012; Kreidberg et al. 2014).”
PASTIS: Bayesian extrasolar planet validation. I. General framework, models, and performance (2014, methodology): “An adaptive step size prescription closely following Ford (2006) was implemented, and the target acceptance rate is set to 25 per cent, since the problems dealt with here are usually multidimensional (see Roberts, Gelman & Gilks 1997, and references therein).”
HAT-P-11b: A SUPER-NEPTUNE PLANET TRANSITING A BRIGHT K STAR IN THE KEPLER FIELD (2009, methodology): “Uncertainties of the parameters were derived using the Markov Chain Monte-Carlo method (MCMC, see Ford 2006), by starting two types of chains from the best fit value.”
The CARMENES Search for Exoplanets around M Dwarfs: A Low-mass Planet in the Temperate Zone of the Nearby K2-18 (2018, background): “The jitter term is treated as an additional source of Gaussian noise with variance σ2jitter and is added in quadrature to the estimated RV uncertainties (Ford 2006).”
THE MASS OF Kepler-93b AND THE COMPOSITION OF TERRESTRIAL PLANETS (2014, methodology): “We increased the efficiency of our model fits by parameterizing eccentric models using √ e cos(ω) and √ e sin(ω) rather than varying e and ω directly (Ford 2006; Eastman et al. 2013).”
Orbital and atmospheric characterization of the planet within the gap of the PDS 70 transition disk (2018, methodology): “To explore the possible orbital solutions of PDS 70 b, we applied the Markov-Chain Monte-Carlo (MCMC) Bayesian analysis technique (Ford 2005, 2006) developed for β Pictoris (Chauvin et al. 2012), and which is well suited for observations covering a small part of the whole orbit (for large orbital…”
The Keck Planet Search: Detectability and the Minimum Mass and Orbital Period Distribution of Extrasolar Planets (2008, background): “Techniques to evaluate the relevant Bayesian integrals over the multidimensional parameter space have been discussed in the literature (Ford 2006; Ford & Gregory 2007; Gregory 2007a, 2007b).”