1932

Abstract

Markov chain Monte Carlo–based Bayesian data analysis has now become the method of choice for analyzing and interpreting data in almost all disciplines of science. In astronomy, over the past decade, we have also seen a steady increase in the number of papers that employ Monte Carlo–based Bayesian analysis. New, efficient Monte Carlo–based methods are continuously being developed and explored. In this review, we first explain the basics of Bayesian theory and discuss how to set up data analysis problems within this framework. Next, we provide an overview of various Monte Carlo–based methods for performing Bayesian data analysis. Finally, we discuss advanced ideas that enable us to tackle complex problems and thus hold great promise for the future. We also distribute downloadable computer software () Python that implements some of the algorithms and examples discussed here.

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2017-08-18
2024-12-07
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Literature Cited

  1. Akaike H. 1974. IEEE Trans. Autom. Control 19:716–23 [Google Scholar]
  2. Andreon S, Weaver B. 2015. Bayesian methods for the physical sciences. Springer Series in Astrostatistics Dordrecht, Neth.: Springer, Cham. [Google Scholar]
  3. Andrieu C, Robert CP. 2001. Controlled MCMC for optimal sampling CiteSeerX. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.2048 [Google Scholar]
  4. Andrieu C, Roberts GO. 2009. Ann. Stat. 37:697–725 [Google Scholar]
  5. Andrieu C, Thoms J. 2008. Stat. Comput. 18:343–73 [Google Scholar]
  6. Barker A. 1965. Aust. J. Phys. 18:119–34 [Google Scholar]
  7. Battaglia G, Helmi A, Morrison H. 2005. MNRAS 364:433 [Google Scholar]
  8. Bayes M, Price M. 1763. Philos. Trans. 53:370–418 [Google Scholar]
  9. Beaumont MA. 2003. Genetics 164:1139–60 [Google Scholar]
  10. Beaumont MA, Zhang W, Balding DJ. 2002. Genetics 162:2025–35 [Google Scholar]
  11. Besag J. 1974. J. R. Stat. Soc. Ser. B (Methodol.) 36:192–236 [Google Scholar]
  12. Binney J. 2011. Pramana 77:39–52 [Google Scholar]
  13. Binney J. 2013. New Astron. Rev. 57:29–51 [Google Scholar]
  14. Binney J, Burnett B, Kordopatis G. et al. 2014. MNRAS 437:351–70 [Google Scholar]
  15. Bland-Hawthorn J, Gerhard O. 2016. Annu. Rev. Astron. Astrophys. 54:529–96 [Google Scholar]
  16. Bovy J. 2016. Ap. J 817:49 [Google Scholar]
  17. Bovy J, Allende Prieto C, Beers TC. et al. 2012a. Ap. J 759:131 [Google Scholar]
  18. Bovy J, Rix HW. 2013. Ap. J 779:115 [Google Scholar]
  19. Bovy J, Rix HW, Liu C. et al. 2012b. Ap. J 753:148 [Google Scholar]
  20. Brewer BJ, Pártay LB, Csányi G. 2011. Stat. Comput. 21:649–56 [Google Scholar]
  21. Brooks S, Gelman A, Jones GL, Meng X-L. 2011. Handbook of Markov Chain Monte Carlo Boca Raton, FL: Chapman and Hall/CRC [Google Scholar]
  22. Brooks SP, Gelman A. 1998. J. Comput. Graph. Stat. 7:434–55 [Google Scholar]
  23. Brown TM, Gilliland RL, Noyes RW, Ramsey LW. 1991. Ap. J 368:599–609 [Google Scholar]
  24. Burnett B, Binney J. 2010. MNRAS 407:339–54 [Google Scholar]
  25. Burnett B, Binney J, Sharma S. 2011. Astron. Astrophys. 532:A113 [Google Scholar]
  26. Burnham KP, Anderson DR. 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach New York: Springer-Verlag, 2nd ed.. [Google Scholar]
  27. Cappé O, Douc R, Guillin A, Marin JM, Robert CP. 2008. Stat. Comput. 18:447–59 [Google Scholar]
  28. Celeux G, Diebolt J. 1985. Comput. Stat. Q. 2:73–82 [Google Scholar]
  29. Chaplin WJ, Kjeldsen H, Christensen-Dalsgaard J. et al. 2011. Science 332:213–16 [Google Scholar]
  30. Chaplin WJ, Miglio A. 2013. Annu. Rev. Astron. Astrophys. 51:353–92 [Google Scholar]
  31. Chen F, Lovász L, Pak I. 1999. Proc. 31st Ann. ACM Symp. Theory Comput., Atlanta, GA May 1–4 275–91 New York: ACM [Google Scholar]
  32. Christen JA, Fox C. 2010. Bayesian Anal. 5:263–81 [Google Scholar]
  33. Christensen N, Meyer R. 1998. Phys. Rev. D 58:082001 [Google Scholar]
  34. Christensen N, Meyer R, Knox L, Luey B. 2001. Class. Quantum Gravity 18:2677–88 [Google Scholar]
  35. Cowles M, Carlin B. 1996. J. Am. Stat. Assoc. 91:883–904 [Google Scholar]
  36. Cox RT. 1946. Am. J. Phys. 14:1–13 [Google Scholar]
  37. Cubillos P, Harrington J, Loredo TJ. 2017. Astron. J. 153:3 [Google Scholar]
  38. de Laplace P. 1774. Mém. Acad. R. Sci. Paris 6:353–71 [Google Scholar]
  39. Deason AJ, Belokurov V, Evans NW. 2012. MNRAS 425:2840 [Google Scholar]
  40. Dempster AP, Laird NM, Rubin DB. 1977. J. R. Stat. Soc. Ser. B (Methodol.) 39:1–38 [Google Scholar]
  41. Diaconis P, Holmes S, Neal RM. 2000. Ann. Appl. Probab. 10:726–52 [Google Scholar]
  42. Dose V. 2002. Tech. Rep. 83, CIPS, MPI für Plasmaphysik, Garching, Ger.
  43. Duane S, Kennedy AD, Pendleton BJ, Roweth D. 1987. Phys. Lett. B 195:216–22 [Google Scholar]
  44. Duvall T Jr., Harvey J. 1986. Seismology of the Sun and the Distant Stars. NATO Adv. Res. Workshop, Cambridge, United Kingdom105–16 Dordrecht, Neth.: Reidel [Google Scholar]
  45. Eadie GM, Harris WE, Widrow LM. 2015. Ap. J. 806:54 [Google Scholar]
  46. Feroz F, Hobson MP, Bridges M. 2009. MNRAS 398:1601–14 [Google Scholar]
  47. Ford EB, Gregory PC. 2007. In Statistical Challenges in Modern Astronomy IV 371:189–205 San Francisco: ASP [Google Scholar]
  48. Foreman-Mackey D, Hogg DW, Lang D, Goodman J. 2013. Publ. Astron. Soc. Pac. 125:306–12 [Google Scholar]
  49. Gelfand AE, Smith AF. 1990. J. Am. Stat. Assoc. 85:398–409 [Google Scholar]
  50. Gelman A, Carlin J, Stern H. et al. 2013. Bayesian Data Analysis. London, UK: Taylor & Francis, 3rd ed.. [Google Scholar]
  51. Gelman A, Hwang J, Vehtari A. 2014. Stat. Comput. 24:997–1016 [Google Scholar]
  52. Gelman A, Roberts G, Gilks W. 1996. Bayesian Stat. 5:599–608 [Google Scholar]
  53. Gelman A, Rubin DB. 1992. Stat. Sci. 7:457–72 [Google Scholar]
  54. Geman S, Geman D. 1984. IEEE Trans. Pattern Anal. Mach. Intell. 6:721–41 [Google Scholar]
  55. Geyer CJ. 1992. Stat. Sci. 7:473–83 [Google Scholar]
  56. Gilks WR, Roberts GO, George EI. 1994. Statistician 43:179–89 [Google Scholar]
  57. Girolami M, Calderhead B. 2011. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 73:123–214 [Google Scholar]
  58. Goodman J, Sokal AD. 1989. Phys. Rev. D 40:2035 [Google Scholar]
  59. Goodman J, Weare J. 2010. Commun. Appl. Math. Comput. Sci. 5:65–80 [Google Scholar]
  60. Green GM, Schlafly EF, Finkbeiner DP. et al. 2014. Ap. J 783:114 [Google Scholar]
  61. Green GM, Schlafly EF, Finkbeiner DP. et al. 2015. Ap. J 810:25 [Google Scholar]
  62. Green PJ. 1995. Biometrika 82:711–32 [Google Scholar]
  63. Gregory P. 2005. Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica® Support Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  64. Griewank A, Walther A. 2008. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation Philadelphia, PA: Siam [Google Scholar]
  65. Gruberbauer M, Kallinger T, Weiss WW, Guenther DB. 2009. Astron. Astrophys. 506:1043–53 [Google Scholar]
  66. Haario H, Saksman E, Tamminen J. 2001. Bernoulli 7:223–42 [Google Scholar]
  67. Hajian A. 2007. Phys. Rev. D 75:083525 [Google Scholar]
  68. Handberg R, Campante TL. 2011. Astron. Astrophys. 527:A56 [Google Scholar]
  69. Hastings WK. 1970. Biometrika 57:97–109 [Google Scholar]
  70. Hobson MP. 2010. Bayesian Methods in Cosmology Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  71. Hogg DW, Bovy J, Lang D. 2010. arXiv:1008.4686
  72. Hogg DW, Myers AD, Bovy J. 2010. Ap. J 725:2166–75 [Google Scholar]
  73. Homan MD, Gelman A. 2014. J. Mach. Learn. Res. 15:1593–623 [Google Scholar]
  74. Jaynes ET. 1957. Phys. Rev. 106:620 [Google Scholar]
  75. Jaynes ET. 1999. Straight line fitting—a Bayesian solution. http://bayes.wustl.edu/etj/articles/leapz.pdf [Google Scholar]
  76. Jaynes ET. 2003. Probability Theory: The Logic of Science Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  77. Jeffreys H. 1939. Theory of Probability Oxford, UK: Oxford Univ. Press [Google Scholar]
  78. Jørgensen BR, Lindegren L. 2005. Astron. Astrophys. 436:127–43 [Google Scholar]
  79. Kafle PR, Sharma S, Lewis GF, Bland-Hawthorn J. 2012. Ap. J. 761:98 [Google Scholar]
  80. Kafle PR, Sharma S, Lewis GF, Bland-Hawthorn J. 2014. Ap. J. 794:59 [Google Scholar]
  81. Kallinger T, Mosser B, Hekker S. et al. 2010. Astron. Astrophys. 522:A1 [Google Scholar]
  82. Kass RE, Raftery AE. 1995. J. Am. Stat. Assoc. 90:773–95 [Google Scholar]
  83. Kass RE, Wasserman L. 1996. J. Am. Stat. Assoc. 91:1343–70 [Google Scholar]
  84. Kilbinger M, Wraith D, Robert CP. et al. 2010. MNRAS 405:2381–90 [Google Scholar]
  85. Kirkpatrick S, Gelatt CD, Vecchi MP. et al. 1983. Science 220:671–80 [Google Scholar]
  86. Kjeldsen H, Bedding TR. 1995. Astron. Astrophys. 293:87–106 [Google Scholar]
  87. Knox L, Christensen N, Skordis C. 2001. Ap. J. Lett. 563:L95–98 [Google Scholar]
  88. Kuijken K, Gilmore G. 1991. Ap. J. Lett. 367:L9–13 [Google Scholar]
  89. Lewis A, Bridle S. 2002. Phys. Rev. D 66:103511 [Google Scholar]
  90. Liang F, Liu C, Carroll R. 2011. Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples 714 Chichester, UK: John Wiley & Sons [Google Scholar]
  91. Liang F, Wong WH. 2001a. J. Am. Stat. Assoc. 96:653–66 [Google Scholar]
  92. Liang F, Wong WH. 2001b. J. Chem. Phys. 115:3374–80 [Google Scholar]
  93. Lindley DV. 1957. Biometrika 44:187–92 [Google Scholar]
  94. Loredo TJ. 1990. Maximum Entropy and Bayesian Methods J Skilling 81–142 Dordrecht, Neth: Springer Sci.+Bus. Media [Google Scholar]
  95. Macciò AV, Dutton AA, van den Bosch FC. 2007. MNRAS 378:55 [Google Scholar]
  96. MacEachern SN, Berliner LM. 1994. Am. Stat. 48:188–90 [Google Scholar]
  97. MacKay DJ. 2003. Information Theory, Inference and Learning Algorithms Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  98. Magorrian J. 2014. MNRAS 437:2230–48 [Google Scholar]
  99. McMillan PJ. 2011. MNRAS 414:2446–57 [Google Scholar]
  100. McMillan PJ. 2017. MNRAS 465:76–94 [Google Scholar]
  101. McMillan PJ, Binney J. 2012. MNRAS 419:2251–63 [Google Scholar]
  102. McMillan PJ, Binney JJ. 2013. MNRAS 433:1411–24 [Google Scholar]
  103. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E. 1953. J. Chem. Phys. 21:1087–92 [Google Scholar]
  104. Metropolis N, Ulam S. 1949. J. Am. Stat. Assoc. 44:335–41 [Google Scholar]
  105. Møller J, Pettitt AN, Reeves R, Berthelsen KK. 2006. Biometrika 93:451–58 [Google Scholar]
  106. Mukherjee P, Parkinson D, Liddle AR. 2006. Ap. J. Lett. 638:L51–54 [Google Scholar]
  107. Müller P. 1991. Tech. Rep. 91-09 Dep. Stat., Purdue Univ. [Google Scholar]
  108. Murray I, Ghahramani Z, MacKay D. 2006. Proc. 22nd Conf. Uncertainty Artif. Intell., Cambridge, Mass., July 13–16359–66 Arlington, VA: AUAI [Google Scholar]
  109. Neal RM. 1993. Tech. Rep. CRG-TR-93-1 Dep. Comput. Sci., Univ Toronto: [Google Scholar]
  110. Neal RM. 2000. J. Comput. Graph. Stat. 9:249–65 [Google Scholar]
  111. Neal RM. 2011. Handb. Markov Chain Monte Carlo S Brooks, A Gelman, GL Jones, X-L Meng 113–62 Boca Raton, FL: Chapman and Hall/CRC [Google Scholar]
  112. Nelson B, Ford EB, Payne MJ. 2014. Ap. J. Suppl. 21011 [Google Scholar]
  113. Nelson BE, Robertson PM, Payne MJ. 2016. MNRAS 4552484–99 [Google Scholar]
  114. Ness M, Hogg DW, Rix HW, Ho AYQ, Zasowski G. 2015. Ap. J 808:16 [Google Scholar]
  115. Parkinson D, Liddle AR. 2013. Stat. Anal. Data Min. 6:3–14 [Google Scholar]
  116. Parkinson D, Mukherjee P, Liddle AR. 2006. Phys. Rev. D 73:123523 [Google Scholar]
  117. Peskun PH. 1973. Biometrika 60:607–12 [Google Scholar]
  118. Piffl T, Binney J, McMillan PJ. et al. 2014. MNRAS 445:3133–51 [Google Scholar]
  119. Pont F, Eyer L. 2004. MNRAS 351:487–504 [Google Scholar]
  120. Press WH. 1997. Unsolved Problems in Astrophysics JN Bahcall, JP Ostriker 49–60 Princeton, NJ: Princeton Univ. Press [Google Scholar]
  121. Raftery AE, Lewis S. 1992. Bayesian Stat. 4:763–73 [Google Scholar]
  122. Reid MJ, Brunthaler A. 2004. Ap. J 616:872–84 [Google Scholar]
  123. Reid MJ, Menten KM, Brunthaler A. et al. 2014. Ap. J 783:130 [Google Scholar]
  124. Reid MJ, Menten KM, Zheng XW. et al. 2009. Ap. J 700:137–48 [Google Scholar]
  125. Rix HW, Bovy J. 2013. Astron. Astrophys. Rev. 21:61 [Google Scholar]
  126. Robbins H, Monro S. 1951. Ann. Math. Stat. 22:400–7 [Google Scholar]
  127. Robert C, Casella G. 2011. Handb. Markov Chain Monte Carlo S Brooks, A Gelman, GL Jones, X-L Meng 49–66 Boca Raton, FL: Chapman and Hall/CRC [Google Scholar]
  128. Robert CP, Casella G. 2004. Monte Carlo Statistical Methods New York: Springer Sci. Bus. Med. 2nd ed. [Google Scholar]
  129. Roberts G, Gilks W. 1994. J. Multivar. Anal. 49:287–98 [Google Scholar]
  130. Saha P, Williams TB. 1994. Astron. J. 107:1295–302 [Google Scholar]
  131. Sale SE. 2012. MNRAS 427:2119–31 [Google Scholar]
  132. Sale SE. 2015. MNRAS 452:2960–72 [Google Scholar]
  133. Sale SE, Magorrian J. 2015. MNRAS 448:1738–50 [Google Scholar]
  134. Sanders JL, Binney J. 2015. MNRAS 449:3479–502 [Google Scholar]
  135. Schönrich R, Binney J, Dehnen W. 2010. MNRAS 403:1829–33 [Google Scholar]
  136. Schwarz G. 1978. Ann. Stat. 6:461–64 [Google Scholar]
  137. Sharma S, Bland-Hawthorn J, Binney J. et al. 2014. Ap. J 793:51 [Google Scholar]
  138. Sharma S, Stello D, Bland-Hawthorn J, Huber D, Bedding TR. 2016. Ap. J 822:15 [Google Scholar]
  139. Sivia D, Skilling J. 2006. Data Analysis: A Bayesian Tutorial Oxford, UK: Oxford Univ. Press [Google Scholar]
  140. Skilling J. 2006. Bayesian Anal. 1:833–59 [Google Scholar]
  141. Smith AF, Spiegelhalter DJ. 1980. J. R. Stat. Soc. Ser. B 42:213–20 [Google Scholar]
  142. Smith MC, Evans NW, Belokurov V. 2009. MNRAS 399:1223 [Google Scholar]
  143. Sokal A. 1997. Functional Integration 361 NATO ASI Series C DeWitt-Morette, P Cartier, A Folacci 131–92 New York: Springer Sci.+Bus. Media [Google Scholar]
  144. Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A. 2002. J. R. Stat. Soc. Ser. B 64:583–639 [Google Scholar]
  145. Stello D, Huber D, Bedding TR. et al. 2013. Ap. J. Lett. 765:L41 [Google Scholar]
  146. Stello D, Huber D, Sharma S. et al. 2015. Ap. J. Lett. 809:L3 [Google Scholar]
  147. Tanner MA, Wong WH. 1987. J. Am. Stat. Assoc. 82:528–40 [Google Scholar]
  148. Tanner MA, Wong WH. 2010. Stat. Sci. 25:506–16 [Google Scholar]
  149. Taylor JF, Ashdown MAJ, Hobson MP. 2008. MNRAS 389:1284–92 [Google Scholar]
  150. Ter Braak CJ. 2006. Stat. Comput. 16:239–49 [Google Scholar]
  151. Trotta R. 2008. Contemp. Phys. 49:71–104 [Google Scholar]
  152. Ulrich RK. 1986. Ap. J. Lett. 306:L37–40 [Google Scholar]
  153. Vehtari A, Ojanen J. 2012. Stat. Surv. 6:142–228 [Google Scholar]
  154. Verdinelli I, Wasserman L. 1995. J. Am. Stat. Assoc. 90:614–18 [Google Scholar]
  155. Wasserstein RL, Lazar NA. 2016. Am. Stat. 702129–33 [Google Scholar]
  156. Watanabe S. 2010. J. Mach. Learn. Res. 11:3571–94 [Google Scholar]
  157. Watanabe S. 2013. J. Mach. Learn. Res. 14:867–97 [Google Scholar]
  158. Weinberg MD. 2012. Bayesian Anal. 7:737–70 [Google Scholar]
  159. Wolfgang A, Rogers LA, Ford EB. 2016. Ap. J. 825:19 [Google Scholar]
  160. Wraith D, Kilbinger M, Benabed K. et al. 2009. Phys. Rev. D 80:023507 [Google Scholar]
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