Promise of Bayesian Inference for Astrophysics

  • Thomas J. Loredo


The “frequentist” approach to statistics, currently dominating statistical practice in astrophysics, is compared to the historically older Bayesian approach, which is now growing in popularity in other scientific disciplines, and which provides unique, optimal solutions to well-posed problems. The two approaches address the same questions with very different calculations, but in simple cases often give the same final results, confusing the issue of whether one is superior to the other. Here frequentist and Bayesian methods are applied to problems where such a mathematical coincidence does not occur, allowing assessment of their relative merits based on their performance, rather than philosophical argument. Emphasis is placed on a key distinction between the two approaches: Bayesian methods, based on comparisons among alternative hypotheses using the single observed data set, consider averages over hypotheses; frequentist methods, in contrast, average over hypothetical alternative data samples and consider hypothesis averaging to be irrelevant. Simple problems are presented that magnify the consequences of this distinction to where common sense can confidently judge between the methods. These demonstrate the irrelevance of sample averaging, and the necessity of hypothesis averaging, revealing frequentist methods to be fundamentally flawed. Bayesian methods are then presented for astrophysically relevant problems using the Poisson distribution, including the analysis of “on/off” measurements of a weak source in a strong background. Weaknesses of the presently used frequentist methods for these problems are straightforwardly overcome using Bayesian methods. Additional existing applications of Bayesian inference to astrophysical problems are noted.


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  1. [1]
    D. Basu. Statistical Information and Likelihood. Sankhyā 37, 1–71. 1975.MATHGoogle Scholar
  2. [2]
    D. Basu. On the Elimination of Nuisance Parameters. J. Amer. Stat. Assoc. 72, 355–366. 1977.MATHCrossRefGoogle Scholar
  3. [3]
    J.O. Berger and D. A. Berry. Statistical Analysis and the Illusion of Objectivity. Amer. Scientist 76. 159. 1988.ADSGoogle Scholar
  4. [4]
    G.L. Bretthorst. Bayesian Spectrum Analysis and Parameter Estimation. Springer-Verlag. New York, 1988.MATHGoogle Scholar
  5. [5]
    G.L. Bretthorst and C.R. Smith. Bayesian Analysis of Signals from Closely-Spaced Objects. In Infrared Systems and Components III (Ed. R.L. Caswell). SPIE 1050. 1989.Google Scholar
  6. [6]
    P. Cheeseman, J. Kelly, M. Self, J. Stutz, W. Taylor, and D. Freeman. AutoClass: A Bayesian Classification System. In Proceedings of the 5th International Conference on Machine Learning (Ed. J. Laird). Morgan Kaufmann. San Mateo. CA, pp. 54–64. 1988.Google Scholar
  7. [7]
    M.L. Cherry E.L. Chupp, P.P. Dunphy, D.J. Forrest, and J.M. Ryan. Statistical Evaluation of Gamma-Ray Line Observations. Astrophys. J. 242. 1257, 1980.ADSCrossRefGoogle Scholar
  8. [8]
    T. Cleveland. The Analysis of Radioactive Decay With a Small Number of Counts by the Method of Maximum Likelihood. Nuc. Instr. Meth. 214, 451–458, 1983.CrossRefGoogle Scholar
  9. [9]
    J. Cornfield. The Bayesian Outlook and Its Application. Biometrics 25, 617–642, 1969.MathSciNetCrossRefGoogle Scholar
  10. [10]
    A.P. Dawid. A Bayesian Look at Nuisance Parameters. In Bayesian Statistics (Eds. J.M. Bernardo, M.H. DeGroot, D.V. Lindley, and A.F.M. Smith). University Press, Valencia, Spain, p. 167, 1980.Google Scholar
  11. [11]
    W.T. Eadie, D. Drijard, F.E. James, M. Roos, and B. Sadoulet. Sta-tistical Methods in Experimental Physics. North-Holland Publishing Company, Amsterdam, 1971.Google Scholar
  12. [12]
    E.D. Feigelson. Statistics in Astronomy. In Encyclopedia of Statistical Science. Supplement Volume (Eds. S. Kotz and N.L. Johnson). Wiley, New York, p. 7, 1989.Google Scholar
  13. [13]
    N. Gehrels. Confidence Limits for Small Numbers of Events in Astro-physical Data. Astrophys. J. 303, 336–346, 1986.ADSCrossRefGoogle Scholar
  14. [14]
    J. Goebel, K. Volk, H. Walker, F. Gerbault, P. Cheeseman, M. Self, J. Stutz, and W. Taylor. A Bayesian Classification of the IRAS LRS Atlas. Astron. Astrophys. 222, L5–L8, 1989.ADSGoogle Scholar
  15. [15]
    P.C. Gregory and T.J. Loredo. A New Method for the Detection of a Periodic Signal of Unknown Shape and Period. Astrophys. J., 1992, in press.Google Scholar
  16. [16]
    S.F. Gull. Bayesian Inductive Inference and Maximum Entropy. In Maximum-Entropy and Bayesian Methods in Science and Engineering, Vol. 1 (Eds. G.J. Erickson and CR. Smith). Kluwer Academic Publishers. Dordrecht, p. 53, 1988.Google Scholar
  17. [17]
    S.F. Gull. Developments in Maximum Entropy Data Analysis. In Maximum-Entropy and Bayesian Methods (Ed. J. Skilling). Kluwer Academic Publishers, Dordrecht, p. 53. 1989.Google Scholar
  18. [18]
    D. Hearn. Consistent Analysis of Gamma-Ray Astronomy Experiments. Nuc. Instr. Meth. 70, 200. 1969.ADSCrossRefGoogle Scholar
  19. [19]
    O. Helene. Upper Limit of Peak Area. Nuc. Instr. Meth. 212, 319–322, 1983.CrossRefGoogle Scholar
  20. [20]
    C. Howson and P. Urbach. Scientific Reasoning: The Bayesian Approach. Open Court Press, LaSalle, IL, 1989.Google Scholar
  21. [21]
    E.T. Jaynes. Prior Probabilities. IEEE Trans. SSC-4, 227, 1968. Reprinted in E.T. Jaynes, Papers on Probability, Statistics, and Statistical Physics (Ed. R.D. Rosenkrantz). D. Reidel, Dordrecht, 1983, p. 114.Google Scholar
  22. [22]
    E.T. Jaynes. Confidence Intervals vs. Bayesian Intervals. In Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science (Eds. W.L. Harper and C.A. Hooker). D. Reidel, Dordrecht, p. 252, 1976. Reprinted in E.T. Jaynes, Papers on Probability, Statistics, and Statistical Physics (Ed. R.D. Rosenkrantz). D. Reidel, Dordrecht, 1983.Google Scholar
  23. [23]
    E.T. Jaynes. Bayesian Spectrum and Chirp Analysis. In Maximum-Entropy and Bayesian Spectral Analysis and Estimation Problems (Eds. C.R. Smith and G.J. Erickson). D. Reidel, Dordrecht, p. 1, 1987.CrossRefGoogle Scholar
  24. [24]
    E.T. Jaynes. Detection of Extra-Solar System Planets. In Maximum-Entropy and Bayesian Methods in Science and Engineering. Vol. 1 (Eds. G.J. Erickson and C.R. Smith). Kluwer Academic Publishers, Dordrecht, p. 147, 1988.Google Scholar
  25. [25]
    H. Jeffreys. Theory of Probability. Oxford Univ. Press, Oxford. 1961 (1st edition 1939).MATHGoogle Scholar
  26. [26]
    R.P. Kraft, D.N. Burrows, and J.A. Nousek. Determination of Confidence Limits for Experiments with Low Numbers of Counts. Astro-phys. J. 374. 344–355, 1991.ADSCrossRefGoogle Scholar
  27. [27]
    M. Lampton, B. Margon, and S. Bowyer. Parameter Estimation in X-Ray Astronomy. Astrophys. J. 208, 177, 1976.ADSCrossRefGoogle Scholar
  28. [28]
    D.A. Leahy, W. Darbro, R.F. Eisner, M.C. Weisskopf, P.G. Sutherland, S. Kahn, and J.E. Grindlay. On Searches for Pulsed Emission with Application to Four Globular Cluster X-Ray Sources: NGC 1851. 6441. 6624, and 6712. Astrophys. J. 266. 160–170, 1983.ADSCrossRefGoogle Scholar
  29. [29]
    T.-P.Li and Y.-Q. Ma. Analysis Methods for Results in Gamma-Ray Astronomy. Astrophys. J. 272, 317–324. 1983.ADSCrossRefGoogle Scholar
  30. [30]
    D.V. Lindley. The 1988 Wald Memorial Lectures: The Present Position in Bayesian Statistics. Statist. Sci. 5, 44–89, 1990.MathSciNetMATHGoogle Scholar
  31. [31]
    D.V. Lindley and Phillips. Inference for a Bernoulli Process (a Bayesian View). Am. Statistician 30, 112–119, 1976.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    J.A. Lobo. Estimation of the Arrival Times of Gravitational Waves From Coalescing Binaries: The Performance of a Long-Baseline Inter-ferometric Gravitational Wave Antenna. Mon. Not. Roy. Astr. Soc. 247, 573–583. 1990.ADSGoogle Scholar
  33. [33]
    T.J. Loredo. Prom Laplace to Supernova SN 1987A: Bayesian Inference in Astrophysics. In Maximum-Entropy and Bayesian Methods (Ed. P. Fougère). Kluwer Academic Publishers, Dordrecht, pp. 81–142, 1990.CrossRefGoogle Scholar
  34. [34]
    T.J. Loredo and D.Q. Lamb. Neutrinos from SN 1987A: Implications for Cooling of the Nascent Neutron Star and the Mass of the Electron Antineutrino. In Proceedings of the 14th Texas Symposium on Relativistic Astrophysics (Ed. E. Fenyves). Ann. N. Y. Acad. Sci. 571, 601, 1989.Google Scholar
  35. [35]
    T.J. Loredo and D.Q. Lamb. Bayesian Analysis of Neutrinos from SN 1987A: Implications for Cooling of the Nascent Neutron Star and for the Rest Mass of the Electron Antineutrino (submitted to Phys. Rev. D.).Google Scholar
  36. [36]
    C.A. Morrow and T.M. Brown. A Bayesian Approach to Ridge Fitting in the ω - k Diagram of the Solar Five-Minute Oscillations. In Advances in Helio- and A steroScismology (Eds. J. Christensen-Dalsgaard and S. Frandsen). Reidel, Dordrecht, Netherlands, pp. 485–489, 1988.CrossRefGoogle Scholar
  37. [37]
    W. L. Nicholson. Statistics of Net-Counting-Rate Estimation with Dominant Background Corrections. Nucleonics 24 (8), 118–121, 1966.Google Scholar
  38. [38]
    E. O’Mongain. Application of Statistics to Results in Gamma Ray Astronomy. Nature 241. 376, 1973.ADSCrossRefGoogle Scholar
  39. [39]
    W.H. Press. B.P. Flannery, S.A. Teukolsky, and W.T. Vetterling. Nu-merical Recipes: The Art of Scientific Computing. Cambridge Univ. Press, Cambridge, 1986.Google Scholar
  40. [40]
    L. J. Rainwater and C.S. Wu. Applications of Probability Theory to Nuclear Particle Detection. Nucleonics 1(2), 60–69, 1947.Google Scholar
  41. [41]
    A. Sard and R.D. Sard. Some Statistical Considerations on Coincidence Counting. Rev. Sci. Instr. 20, 526, 1949.MathSciNetADSCrossRefGoogle Scholar
  42. [42]
    S. Sibisi. Quantified MAXENT: An NMR Application. In Maximum Entropy and Bayesian Methods (Ed. P. Fougere). Kluwer Academic Publishers, Dordrecht, pp. 351–358, 1990.CrossRefGoogle Scholar
  43. [43]
    J. Skilling. Quantified Maximum Entropy. In Maximum. Entropy and Bayesian Methods (Ed. P. Fougere). Kluwer Academic Publishers, Dordrecht, pp. 341–350. 1990.CrossRefGoogle Scholar
  44. [44]
    G. Zech. Upper Limits in Experiments with Background or Measurement Errors. Nucl. Inst. Meth. Phys. Res. A277, 608–610. 1989.ADSCrossRefGoogle Scholar
  45. [45]
    S.N. Zhang and D. Ramsden. Statistical Data Analysis for Gamma-Ray Astronomy. Exp. Astron. 1, 145–163, 1990.ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Thomas J. Loredo
    • 1
  1. 1.Department of AstronomyCornell UniversityIthacaUSA

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