Abstract
Exoplanet research is carried out at the limits of the capabilities of current telescopes and instruments. The studied signals are weak and often embedded in complex systematics from instrumental, telluric, and astrophysical sources. Combining repeated observations of periodic events, simultaneous observations with multiple telescopes, different observation techniques, and existing information from theory and prior research can help to disentangle the systematics from the planetary signals and offers synergistic advantages over analyzing observations separately. Bayesian inference provides a self-consistent statistical framework that addresses both the necessity for complex systematics models and the need to combine prior information and heterogeneous observations. This chapter offers a brief introduction to Bayesian inference in the context of exoplanet research, with focus on time series analysis, and finishes with an overview of a set of freely available programming libraries.
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References
Ambikasaran S (2015) Generalized Rybicki Press algorithm. Numer Linear Algebra Appl 22(6):1102–1114. http://doi.wiley.com/10.1002/nla.2003
Ambikasaran S, Foreman-Mackey D, Greengard L, Hogg DW, O’Neil M (2014) Fast direct methods for Gaussian processes and the analysis of NASA Kepler mission data. http://arxiv.org/abs/1403.6015
Anderson DR, Collier Cameron A, Hellier C et al (2011) WASP-30b: A 61 M Jup Brown Dwarf Transiting A V = 12, F8 STAR. Astrophys J 726(2):L19. http://adsabs.harvard.edu/abs/2011ApJ...726L..19A
Betancourt M (2017) The convergence of Markov chain Monte Carlo methods: from the Metropolis method to Hamiltonian Monte Carlo. http://arxiv.org/abs/1706.01520
Brewer BJ (2014) Inference for trans-dimensional Bayesian models with diffusive nested sampling. http://arxiv.org/abs/1411.3921
Brewer BJ, Foreman-Mackey D (2016) DNest4: diffusive nested sampling in C++ and Python. http://arxiv.org/abs/1606.03757
Burke CJ, McCullough PR, Valenti JA et al (2007) XO-2b: transiting hot Jupiter in a metal-rich common proper motion binary. Astrophys J 671(2):2115–2128. http://stacks.iop.org/0004-637X/671/i=2/a=2115
Carter JA, Winn JN (2009) Parameter estimation from time-series data with correlated errors: a wavelet-based method and its application to transit light curves. ApJ 704(1):51–67. http://stacks.iop.org/0004-637X/704/i=1/a=51?key=crossref.38a235bbafd054bbdf05d00d5c364393
Carter JA, Yee JC, Eastman J, Gaudi BS, Winn JN (2008) Analytic approximations for transit light-curve observables, uncertainties, and covariances. ApJ 689(1):499–512. http://adsabs.harvard.edu/abs/2008ApJ...689..499C
Carter JA, Winn JN, Gilliland R, Holman MJ (2009) Near-infrared transit photometry of the exoplanet HD 149026b. Astrophys J 696(1):241–253. http://arxiv.org/abs/0902.1542
Chopin N, Robert CP (2010) Properties of nested sampling. Biometrika 97(3):741–755. http://biomet.oxfordjournals.org/cgi/doi/10.1093/biomet/asq021
Claret A (2004) A new non-linear limb-darkening law for LTE stellar atmosphere models. III-Sloan filters: calculations for-5. 0 not more than log not more than +1, 2000 K not. A&A 1005:1001–1005. http://www.csa.com/partners/viewrecord.php?requester=gs&collection=TRD&recid=2005091623165SO
Claret A (2008) Testing the limb-darkening coefficients measured from eclipsing binaries. A&A 482(1):259–266. http://www.aanda.org/10.1051/0004-6361:200809370
Claret A (2009) Does the HD 209458 planetary system pose a challenge to the stellar atmosphere models? A&A 506(3):1335–1340. http://www.aanda.org/10.1051/0004-6361/200912423
Claret A, Bloemen S (2011) Gravity and limb-darkening coefficients for the Kepler, CoRoT, Spitzer, uvby, UBVRIJHK, and Sloan photometric systems. A&A 529:A75. http://www.aanda.org/10.1051/0004-6361/201116451
Claret A, Hauschildt PH, Witte S (2012) New limb-darkening coefficients for PHOENIX/1D model atmospheres. Astron Astrophys 546:A14. http://www.aanda.org/10.1051/0004-6361/201219849
Claret A, Dragomir D, Matthews JM (2014) Theoretical gravity and limb-darkening coefficients for the MOST satellite photometric system. A&A 567:A3. http://www.aanda.org/articles/aa/full_html/2014/07/aa23515-14/aa23515-14.html
Clyde MA, Berger JO, Bullard F et al (2007) Current challenges in Bayesian model choice. In: Babu GJ, Feigelson ED (eds) Statistical challenges in modern astronomy IV ASP conference series, vol 371. Proceedings of the conference held 12–15 June 2006 at Pennsylvania State University, in University Park, pp 224–240
Csizmadia S, Pasternacki T, Dreyer C et al (2013) The effect of stellar limb darkening values on the accuracy of the planet radii derived from photometric transit observations. A&A 549:A9. http://www.aanda.org/10.1051/0004-6361/201219888
Czekala I, Mandel KS, Andrews SM et al (2017) Disentangling time series spectra with Gaussian processes: applications to radial velocity analysis. http://arxiv.org/abs/1702.05652
Espinoza N, Jordan A (2015) Limb darkening and exoplanets: testing stellar model atmospheres and identifying biases in transit parameters. MNRAS 450(2):1879–1899. http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stv744
Feroz F, Hobson MP, Bridges M (2009) MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics. Mon Not R Astron Soc 398(4):1601–1614. https://academic.oup.com/mnras/article-lookup/doi/10.1111/j.1365-2966.2009.14548.x
Feroz F, Hobson MP, Cameron E, Pettitt AN (2013) Importance nested sampling and the MultiNest algorithm, pp 1–28. http://arxiv.org/abs/1306.2144
Ford EB (2005) Quantifying the uncertainty in the orbits of extrasolar planets. AJ 129(3):1706–1717. http://stacks.iop.org/1538-3881/129/i=3/a=1706
Ford EB (2006) Improving the efficiency of Markov Chain Monte Carlo for analyzing the orbits of extrasolar planets. ApJ 642(1):505–522. http://stacks.iop.org/0004-637X/642/ i=1/a=505
Ford EB, Street G, Gregory PC (2007) Bayesian model selection and extrasolar planet detection. In: Statistical challenges in modern astronomy IV, vol 371, pp 189–205
Foreman-Mackey D, Hogg DW, Lang D, Goodman J (2013) emcee: the MCMC hammer. Publ Astron Soc Pac 125(925):306–312. http://arxiv.org/abs/1202.3665. http://www.jstor.org/stable/info/10.1086/670067
Foreman-Mackey D, Agol E, Angus R, Ambikasaran S (2017) Fast and scalable Gaussian process modeling with applications to astronomical time series. http://arxiv.org/abs/1703.09710
Gelman A, Rubin D (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472. http://www.jstor.org/stable/2246093
Gelman A, Roberts GO, Gilks WR (1996) Efficient metropolis jumping rules. Bayesian Stat 5: 599–607
Gelman A, Carlin JB, Stern HS et al (2013) Bayesian data analysis, 3rd edn. CRC Press. https://books.google.com.pa/books?id=eSHSBQAAQBAJ
Gibson N, Aigrain S, Roberts SJ et al (2012) A Gaussian process framework for modelling instrumental systematics: application to transmission spectroscopy. Mon Not R Astron Soc 419(3):2683–2694. http://doi.wiley.com/10.1111/j.1365-2966.2011.19915.x. http://mnras.oxfordjournals.org/cgi/doi/10.1111/j.1365-2966.2011.19915.x
Giménez A (2006) Equations for the analysis of the light curves of extra-solar planetary transits. A&A 450(3):1231–1237. http://adsabs.harvard.edu/abs/2006A&A...450.1231G
Goodman J, Weare J (2010) Ensemble samplers with affine invariance. Commun Appl Math Comput Sci 5(1):65–80. http://pjm.math.berkeley.edu/camcos/2010/5-1/p04.xhtml
Gregory PC (2005) Bayesian logical data analysis for the physical sciences. Cambridge University Press, Cambridge
Gunter T, Osborne MA, Garnett R, Hennig P, Roberts SJ (2014) Sampling for inference in probabilistic models with fast Bayesian quadrature. Adv Neural Inf Proc Syst, 2789–2797. http://papers.nips.cc/paper/5483-sampling-for-inference-in-probabilistic-models-with-fast-bayesian-quadrature
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97–109. http://biomet.oxfordjournals.org/content/57/1/97.abstract. https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/57.1.97
Hennig P, Osborne MA, Girolami M (2015) Probabilistic numerics and uncertainty in computations. Proc Math Phys Eng Sci R Soc 471(2179):20150,142. http://rspa.royalsocietypublishing.org/content/471/2179/20150142
Hoffman MD, Gelman A (2011) The no-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo (2008):30. http://arxiv.org/abs/1111.4246
Holman MJ, Winn JN, Latham DW et al (2006) The transit light curve project. I. Four consecutive transits of the exoplanet XO-1b. ApJ 652(2):1715–1723. http://stacks.iop.org/0004-637X/652/i=2/a=1715
Kass RE, Raftery AE (1995) Bayes factors. J Am Stat Assoc 90(430):773–795. http://www.jstor.org/stable/10.2307/2291091
Kipping DM (2010) Investigations of approximate expressions for the transit duration. MNRAS 407(1):301–313. http://arxiv.org/abs/1004.3819
Kipping DM (2013) Efficient, uninformative sampling of limb darkening coefficients for two-parameter laws. Mon Not R Astron Soc 435(3):2152–2160. http://mnras.oxfordjournals.org/cgi/doi/10.1093/mnras/stt1435. http://arxiv.org/abs/1308.0009
Kipping DM (2014) Characterizing distant worlds with asterodensity profiling. Mon Not R Astron Soc 440(3):2164–2184. http://arxiv.org/abs/1311.1170. https://doi.org/10.1093/mnras/stu318. https://academic.oup.com/mnras/article-lookup/doi/10.1093/mnras/stu318
Kipping DM (2016) Efficient, uninformative sampling of limb-darkening coefficients for a three-parameter law. Mon Not R Astron Soc 455(2):1680–1690. https://academic.oup.com/mnras/article-lookup/doi/10.1093/mnras/stv2379
Link WA, Eaton MJ (2012) On thinning of chains in MCMC. Methods Ecol Evol 3(1):112–115
MacKay DJ (2003) Information theory, inference, and learning algorithms, 7th edn. Cambridge University Press, Cambridge. https://doi.org/10.1017/S026357470426043X. http://www.cambridge.org/0521642981. http://www.inference.phy.cam.ac.uk/mackay/itila/. http://www.journals.cambridge.org/abstract_S026357470426043X. http://www.ncbi.nlm.nih.gov/pubmed/13217055
Mandel KS, Agol E (2002) Analytic light curves for planetary transit searches. ApJ 580(2):L171–L175. http://adsabs.harvard.edu/abs/2002ApJ...580L.171M
Matthews AGDG, van der Wilk M, Nickson T et al (2016) GPflow: a Gaussian process library using TensorFlow, 1–6. http://arxiv.org/abs/1610.08733
Müller HM, Huber KF, Czesla S, Wolter U, Schmitt JHMM (2013) High-precision stellar limb-darkening measurements. A&A 560:A112. http://www.aanda.org/10.1051/0004-6361/201322079
Osborne MA, Garnett R, Roberts SJ et al (2012) Bayesian quadrature for ratios. In: Proceedings of the fifteenth international conference on artificial intelligence and statistics, vol 22, pp 832–840. http://jmlr.csail.mit.edu/proceedings/papers/v22/osborne12.html
Parviainen H, Aigrain S (2015) ldtk: limb darkening toolkit. Mon Not R Astron Soc 453(4):3821–3826. http://arxiv.org/abs/1508.02634. https://doi.org/10.1093/mnras/stv1857. http://mnras.oxfordjournals.org/lookup/doi/10.1093/mnras/stv1857
Rajpaul V, Aigrain S, Osborne MA, Reece S, Roberts SJ (2015) A Gaussian process framework for modelling stellar activity signals in radial velocity data. Mon Not R Astron Soc 452(3): 2269–2291
Rasmussen CE, Ghahramani Z (2002) Bayesian Monte Carlo. Advances in neural … (1). http://machinelearning.wustl.edu/mlpapers/paper_files/AA01.pdf
Rasmussen CE, Williams C (2006) Gaussian processes for machine learning. MIT Press, Cambridge. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.3414
Robert CP (2007) The Bayesian choice. Springer, New York
Roberts GO, Gelman A, Gilks WR (1997) Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann Appl Probab 7(1):110–120
Roberts SJ, Osborne MA, Ebden M et al (2013) Gaussian processes for time-series modelling. Philos Trans Ser A Math Phys Eng Sci 371(1984):20110,550. http://www.ncbi.nlm.nih.gov/pubmed/23277607
Salvatier J, Wiecki T, Fonnesbeck C (2015) Probabilistic programming in Python using PyMC. Arxiv, pp 1–24. http://arxiv.org/abs/1507.08050
Seager S, Mallen-Ornelas G (2003) A unique solution of planet and star parameters from an extrasolar planet transit light curve. ApJ 585(2):1038–1055. http://adsabs.harvard.edu/abs/2003ApJ...585.1038S
Shah A, Wilson AG, Ghahramani Z (2014) Student-t processes as alternatives to Gaussian processes 33:13. http://arxiv.org/abs/1402.4306
Sing DK (2010) Stellar limb-darkening coefficients for CoRot and Kepler. A&A 510:A21. http://www.aanda.org/10.1051/0004-6361/200913675
Sing DK, Désert JM, Lecavelier des Etangs A et al (2009) Transit spectrophotometry of the exoplanet HD 189733b. Astron Astrophys 505(2):891–899. http://arxiv.org/abs/0907.4991v1%5Cnpapers3://publication/uuid/2E560D9E-CF3E-44A1-B205-25C2C919A5F8. http://www.aanda.org/10.1051/0004-6361/200912776
Skilling J (2004) Nested sampling. AIP Conf Proc 735:395–405. http://link.aip.org/link/?APC/735/395/1&Agg=doi
Skilling J (2006) Nested sampling for general Bayesian computation. In: ISBA 8th world meeting on Bayesian statistics. http://projecteuclid.org/euclid.ba/1340370944
Tingley B, Bonomo AS, Deeg HJ (2011) Using stellar densities to evaluate transiting exoplanetary candidates. ApJ 726(2):112. http://stacks.iop.org/0004-637X/726/i=2/a=112?key=crossref.1f7718777e8ac3923c916ecf58d816a4
Tran D, Kucukelbir A, Dieng AB et al (2016) Edward: a library for probabilistic modeling, inference, and criticism. http://arxiv.org/abs/1610.09787
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Parviainen, H. (2018). Bayesian Methods for Exoplanet Science. In: Deeg, H., Belmonte, J. (eds) Handbook of Exoplanets . Springer, Cham. https://doi.org/10.1007/978-3-319-55333-7_149
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