Skip to main content

The Jeffreys–Lindley paradox and discovery criteria in high energy physics

An Erratum to this article was published on 19 February 2015

Abstract

The Jeffreys–Lindley paradox displays how the use of a \(p\) value (or number of standard deviations \(z\)) in a frequentist hypothesis test can lead to an inference that is radically different from that of a Bayesian hypothesis test in the form advocated by Harold Jeffreys in the 1930s and common today. The setting is the test of a well-specified null hypothesis (such as the Standard Model of elementary particle physics, possibly with “nuisance parameters”) versus a composite alternative (such as the Standard Model plus a new force of nature of unknown strength). The \(p\) value, as well as the ratio of the likelihood under the null hypothesis to the maximized likelihood under the alternative, can strongly disfavor the null hypothesis, while the Bayesian posterior probability for the null hypothesis can be arbitrarily large. The academic statistics literature contains many impassioned comments on this paradox, yet there is no consensus either on its relevance to scientific communication or on its correct resolution. The paradox is quite relevant to frontier research in high energy physics. This paper is an attempt to explain the situation to both physicists and statisticians, in the hope that further progress can be made.

This is a preview of subscription content, access via your institution.

Fig. 1

References

  1. Aad, G., et al. (2012). Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC. Physics Letters B, 716(1), 1–29. doi:10.1016/j.physletb.2012.08.020.

    Article  Google Scholar 

  2. Aad, G., et al. (2013). Measurements of Higgs boson production and couplings in diboson final states with the ATLAS detector at the LHC. Physics Letters B, 726, 88–119. doi:10.1016/j.physletb.2013.08.010.

    Article  Google Scholar 

  3. Aaij, R., et al. (2013). Measurement of the \(B^0_s \rightarrow \mu ^+ \mu ^-\) branching fraction and search for \(B^0 \rightarrow \mu ^+ \mu ^-\) decays at the LHCb experiment. Physical Review Letters, 111, 101805. doi: 10.1103/PhysRevLett.111.101805.

    Article  Google Scholar 

  4. Aaltonen, T., et al. (2009). First observation of electroweak single top quark production. Physical Review Letters, 103, 092002. doi:10.1103/PhysRevLett.103.092002.

  5. Abazov, V., et al. (2009). Observation of single top quark production. Physical Review Letters, 103(092), 001. doi:10.1103/PhysRevLett.103.092001.

    Google Scholar 

  6. Alvarez, L. (1968). Nobel lecture: Recent developments in particle physics. http://www.nobelprize.org/nobel_prizes/physics/laureates/1968/alvarez-lecture.html. Accessed 28 Feb 2014.

  7. Anderson, P. W. (1992). The Reverend Thomas Bayes, needles in haystacks, and the fifth force. Physics Today, 45(1), 9–11. doi:10.1063/1.2809482.

    Article  Google Scholar 

  8. Andrews, D. W. K. (1994). The large sample correspondence between classical hypothesis tests and Bayesian posterior odds tests. Econometrica, 62(5), 1207–1232. http://www.jstor.org/stable/2951513. Accessed 28 Feb 2014.

  9. APA. (2010). Publication manual of the American Psychological Association (6th ed.). Washington, DC: American Psychological Association.

  10. Arisaka, K., et al. (1993). Improved upper limit on the branching ratio \(B(K^0_L\rightarrow \mu ^\pm e^\mp \). Physical Review Letters, 70, 1049–1052. doi: 10.1103/PhysRevLett.70.1049.

    Article  Google Scholar 

  11. Babu, K., et al. (2013). Baryon number violation, arXiv:1311.5285 [hep-ph].

  12. Baker, S., & Cousins, R. D. (1984). Clarification of the use of chi-square and likelihood functions in fits to histograms. Nuclear Instruments and Methods, 221, 437–442. doi:10.1016/0167-5087(84)90016-4.

    Article  Google Scholar 

  13. Barroso, A., Branco, G., & Bento, M. (1984). \({{\rm K}^0_{L}}\rightarrow \bar{\mu } e\): Can it be observed? Physics Letters B, 134(1–2), 123–127. doi: 10.1016/0370-2693(84)90999-7.

    Article  Google Scholar 

  14. Bartlett, M. S. (1957). A comment on D. V. Lindley’s statistical paradox. Biometrika, 44(3/4), 533–534. http://www.jstor.org/stable/2332888. Accessed 28 Feb 2014.

  15. Bayarri, M. J. (1987). [Testing precise hypotheses]: Comment. Statistical Science, 2(3), 342–344. http://www.jstor.org/stable/2245776. Accessed 28 Feb 2014.

  16. Bayarri, M. J., Berger, J. O., Forte, A., & Garca-Donato, G. (2012). Criteria for Bayesian model choice with application to variable selection. The Annals of Statistics, 40(3), 1550–1577. http://www.jstor.org/stable/41713685. Accessed 28 Feb 2014.

  17. Berger, J. (2008). A comparison of testing methodologies. In H. Prosper, L. Lyons & A. De Roeck (Eds.), Proceedings of PHYSTAT LHC Workshop on Statistical Issues for LHC Physics, 27–29 June 2007, CERN, CERN-2008-001 (pp. 8–19). Geneva, Switzerland:CERN. http://cds.cern.ch/record/1021125. Accessed 28 Feb 2014.

  18. Berger, J. (2011). The Bayesian approach to discovery. In H. B. Prosper & L. Lyons (Eds.), Proceedings of PHYSTAT 2011 Workshop on Statistical Issues Related to Discovery Claims in Search Experiments and Unfolding, 17–20 January 2011, CERN, CERN-2011-006 (pp. 17–26). Geneva, Switzerland: CERN. http://cdsweb.cern.ch/record/1306523. Accessed 28 Feb 2014.

  19. Berger, J. O. (1985). Statistical decision theory and Bayesian analysis (2nd ed.). Springer Series in Statistics, New York: Springer.

  20. Berger, J. O., & Delampady, M. (1987a). Testing precise hypotheses. Statistical Science, 2(3), 317–335. http://www.jstor.org/stable/2245772. Accessed 28 Feb 2014.

  21. Berger, J. O., & Delampady, M. (1987b). [Testing precise hypotheses]: Rejoinder. Statistical Science, 2(3), 348–352. http://www.jstor.org/stable/2245779. Accessed 28 Feb 2014.

  22. Berger, J. O., & Mortera, J. (1991). Interpreting the stars in precise hypothesis testing. International Statistical Review / Revue Internationale de Statistique, 59(3), 337–353. http://www.jstor.org/stable/1403691. Accessed 28 Feb 2014.

  23. Berger, J. O., & Pericchi, L. R. (2001). Objective Bayesian methods for model selection: Introduction and comparison. Lecture Notes-Monograph Series, 38, 135–207. http://www.jstor.org/stable/4356165. Accessed 28 Feb 2014.

  24. Berger, J. O., & Sellke, T. (1987). Testing a point null hypothesis: The irreconcilability of p values and evidence. Journal of the American Statistical Association, 82(397), 112–122. http://www.jstor.org/stable/2289131. Accessed 28 Feb 2014.

  25. Berkson, J. (1938). Some difficulties of interpretation encountered in the application of the Chi-square test. Journal of the American Statistical Association, 33(203), 526–536. http://www.jstor.org/stable/2279690. Accessed 28 Feb 2014.

  26. Bernardo, J. M. (1999). Nested hypothesis testing: The Bayesian reference criterion. In J. M. Bernardo, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian statistics 6. Proceedings of the Sixth Valencia International Meeting (pp. 101–130). Oxford, UK: Oxford Universiy Press.

  27. Bernardo, J. M. (2009). [Harold Jeffreys’s theory of probability revisited]: Comment. Statistical Science, 24(2), 173–175. http://www.jstor.org/stable/25681292. Accessed 28 Feb 2014.

  28. Bernardo, J. M. (2011a). Bayes and discovery: Objective Bayesian hypothesis testing. In H. B. Prosper & L. Lyons (Eds.), Proceedings of PHYSTAT 2011 Workshop on Statistical Issues Related to Discovery Claims in Search Experiments and Unfolding, 17–20 January 2011, CERN-2011-006 (pp. 27–49). Geneva, Switzerland: CERN. http://cdsweb.cern.ch/record/1306523. Accessed 28 Feb 2014.

  29. Bernardo, J. M. (2011b). Integrated objective Bayesian estimation and hypothesis testing. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith & M. West (Eds.), Bayesian Statistics 9. Proceedings of the Ninth Valencia International Meeting (pp. 1–68). Oxford, UK: Oxford U. Press. http://www.uv.es/bernardo/. Accessed 28 Feb 2014.

  30. Bernardo, J. M., & Rueda, R. (2002). Bayesian hypothesis testing: A reference approach. International Statistical Review / Revue Internationale de Statistique, 70(3), 351–372. http://www.jstor.org/stable/1403862. Accessed 28 Feb 2014.

  31. Box, G. E. P. (1976). Science and statistics. Journal of the American Statistical Association, 71(356), 791–799. http://www.jstor.org/stable/2286841. Accessed 28 Feb 2014.

  32. Box, G. E. P. (1980). Sampling and Bayes’ inference in scientific modelling and robustness. Journal of the Royal Statistical Society Series A (General), 143(4), 383–430. http://www.jstor.org/stable/2982063. Accessed 28 Feb 2014.

  33. Casella, G., & Berger, R. L. (1987a). Reconciling Bayesian and frequentist evidence in the one-sided testing problem. Journal of the American Statistical Association, 82(397), 106–111. http://www.jstor.org/stable/2289130. Accessed 28 Feb 2014.

  34. Casella, G., & Berger, R. L. (1987b). [Testing precise hypotheses]: Comment. Statistical Science, 2(3), 344–347. http://www.jstor.org/stable/2245777. Accessed 28 Feb 2014.

  35. CERN (2013). CERN experiments put Standard Model to stringent test. http://press.web.cern.ch/press-releases/2013/07/cern-experiments-put-standard-model-stringent-test. Accessed 28 Feb 2014.

  36. Chatrchyan, S., et al. (2012). Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Physics Letters B, 716(1), 30–61. doi:10.1016/j.physletb.2012.08.021.

    Article  Google Scholar 

  37. Chatrchyan, S., et al. (2013a). Measurement of the \(B^0_s \rightarrow \mu ^+ \mu ^-\) branching fraction and search for \(B^0 \rightarrow \mu ^+ \mu ^-\) with the CMS experiment. Physical Review Letters, 111, 101804. doi: 10.1103/PhysRevLett.111.101804.

    Article  Google Scholar 

  38. Chatrchyan, S., et al. (2013b). Measurement of the properties of a Higgs boson in the four-lepton final state. Physical Review D, 89(2014), 092007. doi:10.1103/PhysRevD.89.092007.

  39. Chatrchyan, S., et al. (2013c). Study of the mass and spin-parity of the Higgs boson candidate via its decays to \(Z\) boson pairs. Physical Review Letters, 110, 081803. doi: 10.1103/PhysRevLett.110.081803.

    Article  Google Scholar 

  40. Cousins, R. D. (2005). Treatment of nuisance parameters in high energy physics, and possible justifications and improvements in the statistics literature. In L. Lyons & M. K. Unel (Eds.), Proceedings of PHYSTAT 05 Statistical Problems in Particle Physics, Astrophysics and Cosmology, September 12–15, 2005 (pp. 75–85). Oxford: Imperial College Press. http://www.physics.ox.ac.uk/phystat05/proceedings/. Accessed 28 Feb 2014.

  41. Cousins, R. D., & Highland, V. L. (1992). Incorporating systematic uncertainties into an upper limit. Nuclear Instruments and Methods A, 320, 331–335. doi:10.1016/0168-9002(92)90794-5.

    Article  Google Scholar 

  42. Cowan, G., Cranmer, K., Gross, E., & Vitells, O. (2011). Asymptotic formulae for likelihood-based tests of new physics. European Physical Journal C, 71, 1554. doi:10.1140/epjc/s10052-011-1554-0.

    Article  Google Scholar 

  43. Cox, D. R. (2006). Principles of statistical inference. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  44. Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika, 74(1), 33–43.

    Google Scholar 

  45. Demortier, L. (2011). Open issues in the wake of Banff 2010. In: H. B. Prosper & L. Lyons (Eds.), Proceedings of PHYSTAT 2011 Workshop on Statistical Issues Related to Discovery Claims in Search Experiments and Unfolding, 17–20 January 2011, CERN-2011-006, (pp. 1–11). Geneva, Switzerland: CERN. http://cdsweb.cern.ch/record/1306523. Accessed 28 Feb 2014.

  46. Dickey, J. M. (1977). Is the tail area useful as an approximate Bayes factor? Journal of the American Statistical Association, 72(357), 138–142. doi:10.1080/01621459.1977.10479922, http://www.jstor.org/stable/2286921. Accessed 28 Feb 2014.

  47. Eadie, W., et al. (1971). Statistical methods in experimental physics (1st ed.). Amsterdam: North Holland.

    Google Scholar 

  48. Edwards, W., Lindman, H., & Savage, L. J. (1963). Bayesian statistical inference for psychological research. Psychological Review, 70(3), 193–242.

    Article  Google Scholar 

  49. Feldman, G. J., & Cousins, R. D. (1998). Unified approach to the classical statistical analysis of small signals. Phys Rev D, 57, 3873–3889. doi:10.1103/PhysRevD.57.3873, physics/9711021.

  50. Ferguson, C. J., & Heene, M. (2012). A vast graveyard of undead theories: Publication bias and psychological science’s aversion to the null. Perspectives on Psychological Science, 7(6), 555–561. doi:10.1177/1745691612459059.

    Article  Google Scholar 

  51. Fermilab. (2009). Fermilab collider experiments discover rare single top quark. http://www.fnal.gov/pub/presspass/press_releases/Single-Top-Quark-March2009.html. Accessed 28 Feb 2014.

  52. Galison, P. (1983). How the first neutral-current experiments ended. Reviews of Modern Physics, 55, 477–509. doi:10.1103/RevModPhys.55.477.

    Article  Google Scholar 

  53. Gelman, A., & Rubin, D. B. (1995). Avoiding model selection in Bayesian social research. Sociological Methodology, 25, 165–173. http://www.jstor.org/stable/271064. Accessed 28 Feb 2014.

  54. Georgi, H. (1993). Effective field theory. Annual Review of Nuclear and Particle Science, 43, 209–252. doi:10.1146/annurev.ns.43.120193.001233.

    Article  Google Scholar 

  55. Good, I. J. (1992). The Bayes/non-Bayes compromise: A brief review. Journal of the American Statistical Association, 87(419), 597–606. http://www.jstor.org/stable/2290192. Accessed 28 Feb 2014.

  56. Gross, E., & Vitells, O. (2010). Trial factors or the look elsewhere effect in high energy physics. European Physical Journal C, 70, 525–530. doi:10.1140/epjc/s10052-010-1470-8.

    Article  Google Scholar 

  57. Hasert, F., et al. (1973). Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment. Physics Letters B, 46(1), 138–140. doi:10.1016/0370-2693(73)90499-1.

    Article  Google Scholar 

  58. Hirsch, M., Päs, H., & Porod, W. (2013). Ghostly beacons of new physics. Scientific American, 308(April), 40–47. doi:10.1038/scientificamerican0413-40.

    Article  Google Scholar 

  59. Incandela, J., & Gianotti, F. (2012). Latest update in the search for the Higgs boson, public seminar at CERN. Video: http://cds.cern.ch/record/1459565; slides: http://indico.cern.ch/conferenceDisplay.py?confId=197461. Accessed 28 Feb 2014.

  60. James, F. (1980). Interpretation of the shape of the likelihood function around its minimum. Computer Physics Communications, 20, 29–35. doi:10.1016/0010-4655(80)90103-4.

    Article  Google Scholar 

  61. James, F. (2006). Statistical methods in experimental physics (2nd ed.). Singapore: World Scientific.

    Book  Google Scholar 

  62. Jaynes, E. (2003). Probability theory: The logic of science. Cambridge, UK: Cambridge University Press.

    Book  Google Scholar 

  63. Jeffreys, H. (1961). Theory of probability (3rd ed.). Oxford: Oxford University Press.

    Google Scholar 

  64. Johnstone, D., & Lindley, D. (1995). Bayesian inference given data ‘significant at \(\alpha \)’: Tests of point hypotheses. Theory and Decision, 38(1), 51–60. doi: 10.1007/BF01083168.

    Article  Google Scholar 

  65. Kadane, J. B. (1987). [Testing precise hypotheses]: Comment. Statistical Science, 2(3), 347–348. http://www.jstor.org/stable/2245778. Accessed 28 Feb 2014.

  66. Kass, R. (2009). Comment: The importance of Jeffreys’s legacy. Statistical Science, 24(2), 179–182. http://www.jstor.org/stable/25681294. Accessed 28 Feb 2014.

  67. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795. http://www.jstor.org/stable/2291091. Accessed 28 Feb 2014.

  68. Kass, R. E., & Wasserman, L. (1995). A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. Journal of the American Statistical Association, 90(431), 928–934. http://www.jstor.org/stable/2291327. Accessed 28 Feb 2014.

  69. Kirk, R. E. (1996). Practical significance: A concept whose time has come. Educational and Psychological Measurement, 56(5), 746–759. doi:10.1177/0013164496056005002.

    Article  Google Scholar 

  70. Leamer, E. E. (1978). Specification searches: Ad hoc inference with nonexperimental data. Wiley series in probability and mathematical statistics, New York: Wiley.

  71. Lee, P. M. (2004). Bayesian statistics: An introduction (3rd ed.). Chichester, UK: Wiley.

    Google Scholar 

  72. Lehmann, E., & Romero, J. P. (2005). Testing statistical hypotheses (3rd ed.). New York: Springer.

    Google Scholar 

  73. Lindley, D. (2009). [Harold Jeffreys’s theory of probability revisited]: Comment. Statistical Science, 24(2), 183–184. http://www.jstor.org/stable/25681295. Accessed 28 Feb 2014.

  74. Lindley, D. V. (1957). A statistical paradox. Biometrika, 44(1/2), 187–192. http://www.jstor.org/stable/2333251. Accessed 28 Feb 2014.

  75. Lyons, L. (2010). Comments on ‘look elsewhere effect’. http://www.physics.ox.ac.uk/Users/lyons/LEE_feb7_2010.pdf. Accessed 28 Feb 2014.

  76. Lyons, L. (2013). Discovering the significance of 5 sigma, arXiv:1320.1284.

  77. Mayo, D. G., & Spanos, A. (2006). Severe testing as a basic concept in a Neyman-Pearson philosophy of induction. The British Journal for the Philosophy of Science, 57(2), 323–357. http://www.jstor.org/stable/3873470. Accessed 28 Feb 2014.

  78. Nakagawa, S., & Cuthill, I. C. (2007). Effect size, confidence interval and statistical significance: a practical guide for biologists. Biological Reviews, 82(4), 591–605. doi:10.1111/j.1469-185X.2007.00027.x.

    Article  Google Scholar 

  79. Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences, 236(767), pp. 333–380. http://www.jstor.org/stable/91337. Accessed 28 Feb 2014.

  80. Neyman, J., & Pearson, E. S. (1933a). On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London Series A, Containing Papers of a Mathematical or Physical Character, 231, 289–337. http://www.jstor.org/stable/91247. Accessed 28 Feb 2014.

  81. Neyman, J., & Pearson, E. S. (1933b). The testing of statistical hypotheses in relation to probabilities a priori. Mathematical Proceedings of the Cambridge Philosophical Society, 29, 492–510. doi:10.1017/S030500410001152X.

    Article  Google Scholar 

  82. Philippe, A., & Robert, C. (1998). A note on the confidence properties of reference priors for the calibration model. Sociedad de Estadística e Investigación Operativa Test, 7(1), 147–160. doi:10.1007/BF02565107.

    Google Scholar 

  83. Prescott, C., et al. (1978). Parity non-conservation in inelastic electron scattering. Physics Letters B, 77(3), 347–352. doi:10.1016/0370-2693(78)90722-0.

    Article  Google Scholar 

  84. Raftery, A. E. (1995a). Bayesian model selection in social research. Sociological Methodology, 25, 111–163. http://www.jstor.org/stable/271063. Accessed 28 Feb 2014.

  85. Raftery, A. E. (1995b). Rejoinder: Model selection is unavoidable in social research. Sociological Methodology, 25, 185–195. http://www.jstor.org/stable/271066. Accessed 28 Feb 2014.

  86. Rice, J. A. (2007). Mathematical statistics and data analysis (3rd ed.). Belmont, CA: Thomson.

    Google Scholar 

  87. Robert, C. P. (1993). A note on Jeffreys-Lindley paradox. Statistica Sinica, 3(2), 601–608.

    Google Scholar 

  88. Robert, C.P. (2013). On the Jeffreys-Lindley paradox, arXiv:1303.5973v3 [stat.ME].

  89. Robert, C. P., Chopin, N., & Rousseau, J. (2009). Harold Jeffreys’s theory of probability revisited. Statistical Science, 24(2), 141–172. http://www.jstor.org/stable/25681291. Accessed 28 Feb 2014.

  90. Rosenfeld, A.H. (1968). Are there any far-out mesons or baryons? In: Baltay, C., Rosenfeld, A.H. (eds) Meson spectroscopy: A collection of articles, W.A. Benjamin, New York, pp 455–483, From the preface: based on reviews presented at the Conference on Meson Spectroscopy, April 26–27, 1968, Philadelphia, PA USA. “...not, however, intended to be the proceedings...”.

  91. Senn, S. (2001). Two cheers for p-values? Journal of Epidemiology and Biostatistics, 6(2), 193–204.

  92. Shafer, G. (1982). Lindley’s paradox. Journal of the American Statistical Association, 77(378), 325–334. http://www.jstor.org/stable/2287244. Accessed 28 Feb 2014.

  93. Smith, A. F. M., & Spiegelhalter, D. J. (1980). Bayes factors and choice criteria for linear models. Journal of the Royal Statistical Society Series B (Methodological), 42(2), 213–220. http://www.jstor.org/stable/2984964. Accessed 28 Feb 2014.

  94. Spanos, A. (2013). Who should be afraid of the Jeffreys-Lindley paradox? Philosophy of Science, 80(1), 73–93. http://www.jstor.org/stable/10.1086/668875. Accessed 28 Feb 2014.

  95. Stuart, A., Ord, K., & Arnold, S. (1999). Kendall’s advanced theory of statistics (6th edn., Vol 2A). London: Arnold, and earlier editions by Kendall and Stuart.

  96. Swedish Academy (2013) Advanced information: Scientific background: The BEH-mechanism, interactions with short range forces and scalar particles. http://www.nobelprize.org/nobel_prizes/physics/laureates/2013/advanced.html. Accessed 28 Feb 2014.

  97. ’t Hooft, G. (1976). Symmetry breaking through Bell-Jackiw anomalies. Physical Review Letters, 37, 8–11. doi:10.1103/PhysRevLett.37.8.

    Article  Google Scholar 

  98. ’t Hooft, G. (1999). Nobel lecture: A confrontation with infinity. http://www.nobelprize.org/nobel_prizes/physics/laureates/1999/thooft-lecture.html, This web page has video, slides, and pdf writeup. Accessed 28 Feb 2014.

  99. Thompson, B. (2007). The nature of statistical evidence. Lecture notes in statistics. New York: Springer.

    Google Scholar 

  100. van Dyk, D. A. (2014). The role of statistics in the discovery of a Higgs boson. Annual Review of Statistics and Its Application, 1(1), 41–59. doi:10.1146/annurev-statistics-062713-085841.

    Article  Google Scholar 

  101. Vardeman, S. B. (1987). [Testing a point null hypothesis: The irreconcilability of p values and evidence]: Comment. Journal of the American Statistical Association, 82(397), 130–131. http://www.jstor.org/stable/2289136. Accessed 28 Feb 2014.

  102. Webster (1969). Webster’s Seventh New Collegiate Dictionary, based on Webster’s Third New International Dictionary. MA: G. and C. Merriam: Springfield.

  103. Welch, B. L., & Peers, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihoods. Journal of the Royal Statistical Society Series B (Methodological), 25(2), 318–329. http://www.jstor.org/stable/2984298. Accessed 28 Feb 2014.

  104. Wilczek, F. (2004). Nobel lecture: Asymptotic freedom: From paradox to paradigm. http://www.nobelprize.org/nobel_prizes/physics/laureates/2004/wilczek-lecture.html, This web page has video, slides, and pdf writeup. Accessed 28 Feb 2014.

  105. Wilkinson, L., et al. (1999). Statistical methods in psychology journals—guidelines and explanations. American Psychologist, 54(8), 594–604. doi:10.1037//0003-066X.54.8.594.

    Article  Google Scholar 

  106. Zellner, A. (2009). [Harold Jeffreys’s theory of probability revisited]: Comment. Statistical Science, 24(2), 187–190. http://www.jstor.org/stable/25681297. Accessed 28 Feb 2014.

  107. Zellner, A., & Siow, A. (1980). Posterior odds ratios for selected regression hypotheses. Trabajos de Estadistica Y de Investigacion Operativa, 31(1), 585–603. doi:10.1007/BF02888369.

    Article  Google Scholar 

Download references

Acknowledgments

I thank my colleagues in high energy physics, and in the CMS collaboration in particular, for many useful discussions. I am grateful to members of the CMS Statistics Committee, for comments on an early draft of the manuscript, and in particular to Luc Demortier and Louis Lyons for continued discussions. Tom Ferbel provided invaluable detailed comments on two previous versions that I posted on the arXiv. The PhyStat series of workshops organized by Louis Lyons has led to many fruitful discussions and enlightening contact with prominent members of the statistics community. This material is based upon work partially supported by the U.S. Department of Energy under Award Number DE-SC0009937.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Robert D. Cousins.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cousins, R.D. The Jeffreys–Lindley paradox and discovery criteria in high energy physics. Synthese 194, 395–432 (2017). https://doi.org/10.1007/s11229-014-0525-z

Download citation

Keywords

  • Jeffreys–Lindley paradox
  • Lindley’s paradox
  • Bayesian model selection
  • p values
  • High energy physics