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Extra-solar Planets via Bayesian Fusion MCMC

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Astrostatistical Challenges for the New Astronomy

Part of the book series: Springer Series in Astrostatistics ((SSIA,volume 1))

Abstract

A Bayesian multi-planet Kepler periodogram has been developed based on a fusion Markov chain Monte Carlo algorithm (FMCMC). FMCMC is a new general purpose tool for nonlinear model fitting. It incorporates parallel tempering, simulated annealing and genetic crossover operations. Each of these features facilitate the detection of a global minimum in chi-squared in a highly multi-modal environment. By combining all three, the algorithm greatly increases the probability of realizing this goal.

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Notes

  1. 1.

    For multiple planet models, there is no analytic expression for the exact radial velocity perturbation. In many cases, the radial velocity perturbation can be well modeled as the sum of multiple independent Keplerian orbits which is what has been assumed in this paper.

  2. 2.

    Following on from the pioneering work on Bayesian periodograms by [18, 19]

  3. 3.

    The interval between tempering swap operations is typically much smaller than is suggested by this schematic.

  4. 4.

    Mathematica code that implements the version of fusion MCMC shown in Fig. 7.4 is available on the Cambridge University Press web site for my textbook [10], ‘Bayesian Logical data Analysis for the Physical Sciences’. See the ‘Additional book examples with Mathematica 8 tutorial’ in the resource material. There you will find an example entitled, ‘Markov chain Monte Carlo powered Kepler periodogram’. Non Mathematica users can download a free Wolfram CDF Player to view the resource material.

  5. 5.

    Thinning by a factor of 10 has already occurred meaning only every tenth iteration is recorded.

  6. 6.

    In the absence of detailed knowledge of the sampling distribution for the extra noise, we pick a Gaussian because for any given finite noise variance it is the distribution with the largest uncertainty as measured by the entropy, i.e., the maximum entropy distribution [30] and [10] (section 8.7.4.)

  7. 7.

    Test that the extended credible region (like 9930) for the period parameter does not overlap the credible region of an adjacent period parameter in a multiple planet fit.

  8. 8.

    Table 7.1 gives two different choices of prior for the eccentricity parameter. The marginal likelihoods listed in the Table 7.2 correspond to the eccentricity noise bias prior. Marginal likelihood values assuming a uniform eccentricity prior were systematically lower. For example, for a 3 planet fit using the uniform eccentricity prior the marginal likelihood was a factor of 3 smaller.

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Acknowledgments

The author would like to thank Wolfram Research for providing a complementary license to run gridMathematica.

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  1. Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z1, Canada

    Philip C. Gregory

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    Joseph M. Hilbe

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Gregory, P.C. (2013). Extra-solar Planets via Bayesian Fusion MCMC. In: Hilbe, J. (eds) Astrostatistical Challenges for the New Astronomy. Springer Series in Astrostatistics, vol 1. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3508-2_7

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