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Register a new userApplication of Bayesian model inadequacy criterion for multiple data sets to radial velocity models of exoplanet systems
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We present a simple mathematical criterion for determining whether a given statistical model does not describe several independent sets of measurements, or data modes, adequately. We derive this criterion for two data sets and generalise it to several sets by using the Bayesian updating of the posterior probability density. To demonstrate the usage of the criterion, we apply it to observations of exoplanet host stars by re-analysing the radial velocities of HD 217107, Gliese 581, and u{psion} Andromedae and show that the currently used models are not necessarily adequate in describing the properties of these measurements. We show that while the two data sets of Gliese 581 can be modelled reasonably well, the noise model of HD 217107 needs to be revised. We also reveal some biases in the radial velocities of u{psion} Andromedae and report updated orbital parameters for the recently proposed 4-planet model. Because of the generality of our criterion, no assumptions are needed on the nature of the measurements, models, or model parameters. The method we propose can be applied to any astronomical problems, as well as outside the field of astronomy, because it is a simple consequence of the Bayes rule of conditional probabilities.
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We conducted speckle imaging observations of 53 stellar systems that were members of long-term radial velocity (RV) monitoring campaigns and exhibited substantial accelerations indicative of planetary or stellar companions in wide orbits. Our observations were made with blue and red filters using the Differential Speckle Survey Instrument at Gemini-South and the NN-Explore Exoplanet Stellar Speckle Imager at the WIYN telescope. The speckle imaging identifies eight luminous companions within two arcseconds of the primary stars. In three of these systems (HD 1388, HD 87359, and HD 104304), the properties of the imaged companion are consistent with the RV measurements, suggesting that these companions may be associated with the primary and the cause of the RV variation. For all 53 stellar systems, we derive differential magnitude limits (i.e., contrast curves) from the imaging. We extend this analysis to include upper limits on companion mass in systems without imaging detections. In 25 systems, we rule out companions with mass greater than 0.2 $M_{odot}$, suggesting that the observed RV signals are caused by late M dwarfs or substellar (potentially planetary) objects. On the other hand, the joint RV and imaging analysis almost entirely rules out planetary explanations of the RV signal for HD 19522 and suggests that the companion must have an angular separation below a few tenths of an arcsecond. This work highlights the importance of combined RV and imaging observations for characterizing the outer regions of nearby planetary systems.
We present results from a data challenge posed to the radial velocity (RV) community: namely, to quantify the Bayesian evidence for n={0,1,2,3} planets in a set of synthetically generated RV datasets containing a range of planet signals. Participating teams were provided the same likelihood function and set of priors to use in their analysis. They applied a variety of methods to estimate Z, the marginal likelihood for each n-planet model, including cross-validation, the Laplace approximation, importance sampling, and nested sampling. We found the dispersion in Z across different methods grew with increasing n-planet models: ~3 for 0-planets, ~10 for 1-planet, ~100-1000 for 2-planets, and >10,000 for 3-planets. Most internal estimates of uncertainty in Z for individual methods significantly underestimated the observed dispersion across all methods. Methods that adopted a Monte Carlo approach by comparing estimates from multiple runs yielded plausible uncertainties. Finally, two classes of numerical algorithms (those based on importance and nested samplers) arrived at similar conclusions regarding the ratio of Zs for n and (n+1)-planet models. One analytic method (the Laplace approximation) demonstrated comparable performance. We express both optimism and caution: we demonstrate that it is practical to perform rigorous Bayesian model comparison for <=3-planet models, yet robust planet discoveries require researchers to better understand the uncertainty in Z and its connections to model selection.
GJ667C is the least massive component of a triple star system which lies at a distance of about 6.8 pc (22.1 light-years) from Earth. GJ667C has received much attention recently due to the claims that it hosts up to seven planets including three super-Earths inside the habitable zone. We present a Bayesian technique for the analysis of radial velocity (RV) data-sets in the presence of correlated noise component (red noise), with unknown parameters. We also introduce hyper-parameters in our model in order to deal statistically with under or over-estimated error bars on measured RVs as well as inconsistencies between different data-sets. By applying this method to the RV data-set of GJ667C, we show that this data-set contains a significant correlated (red) noise component with correlation timescale for HARPS data of order 9 days. Our analysis shows that the data only provides strong evidence for the presence of two planets: GJ667Cb and c with periods 7.19d and 28.13d respectively, with some hints towards the presence of a third signal with period 91d. The planetary nature of this third signal is not clear and additional RV observations are required for its confirmation. Previous claims of the detection of additional planets in this system are due the erroneous assumption of white noise. Using the standard white noise assumption, our method leads to the detection of up to five signals in this system. We also find that with the red noise model, the measurement uncertainties from HARPS for this system are under-estimated at the level of ~50 per cent.
Multiple systems estimation is a key approach for quantifying hidden populations such as the number of victims of modern slavery. The UK Government published an estimate of 10,000 to 13,000 victims, constructed by the present author, as part of the strategy leading to the Modern Slavery Act 2015. This estimate was obtained by a stepwise multiple systems method based on six lists. Further investigation shows that a small proportion of the possible models give rather different answers, and that other model fitting approaches may choose one of these. Three data sets collected in the field of modern slavery, together with a data set about the death toll in the Kosovo conflict, are used to investigate the stability and robustness of various multiple systems estimate methods. The crucial aspect is the way that interactions between lists are modelled, because these can substantially affect the results. Model selection and Bayesian approaches are considered in detail, in particular to assess their stability and robustness when applied to real modern slavery data. A new Markov Chain Monte Carlo Bayesian approach is developed; overall, this gives robust and stable results at least for the examples considered. The software and datasets are freely and publicly available to facilitate wider implementation and further research.
A number of giant planet pairs discovered by the radial velocity method with period ratios $lesssim 2$ may reside in mean motion resonances. Convergent orbital migration and resonant capture at the time of formation would naturally explain the present-day resonant orbital configurations of these systems. Planets that experience smooth migration and eccentricity damping forces due to a proto-planetary disk should not only capture into mean motion resonances but also end up in a specific dynamical configuration within the resonance, sometimes referred to as apsidal corotation resonance (ACR). Here we develop a method for testing the hypothesis that a planet pair resides in an ACR by directly fitting radial velocity data. The ACR hypothesis strongly restricts the number of free parameters describing the radial velocity signal and we compare fits using this highly restricted model to fits using a more conventional two-planet RV model by using nested sampling simulations. We apply our method to HD 45364 and HD 33844, two systems hosting giant planet pairs in 3:2 and 5:3 resonances, respectively. We demonstrate that the observations of both systems support an ACR configuration and we use the results of our ACR model fits to constrain possible migration histories of these systems.
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