Statistical Science

J. B. S. Haldane’s Contribution to the Bayes Factor Hypothesis Test

Alexander Etz and Eric-Jan Wagenmakers

Full-text: Open access


This article brings attention to some historical developments that gave rise to the Bayes factor for testing a point null hypothesis against a composite alternative. In line with current thinking, we find that the conceptual innovation—to assign prior mass to a general law—is due to a series of three articles by Dorothy Wrinch and Sir Harold Jeffreys (1919, 1921, 1923a). However, our historical investigation also suggests that in 1932, J. B. S. Haldane made an important contribution to the development of the Bayes factor by proposing the use of a mixture prior comprising a point mass and a continuous probability density. Jeffreys was aware of Haldane’s work and it may have inspired him to pursue a more concrete statistical implementation for his conceptual ideas. It thus appears that Haldane may have played a much bigger role in the statistical development of the Bayes factor than has hitherto been assumed.

Article information

Statist. Sci. Volume 32, Number 2 (2017), 313-329.

First available in Project Euclid: 11 May 2017

Permanent link to this document

Digital Object Identifier

History of statistics induction evidence Sir Harold Jeffreys


Etz, Alexander; Wagenmakers, Eric-Jan. J. B. S. Haldane’s Contribution to the Bayes Factor Hypothesis Test. Statist. Sci. 32 (2017), no. 2, 313--329. doi:10.1214/16-STS599.

Export citation


  • Aldrich, J. (2005). The statistical education of Harold Jeffreys. Int. Stat. Rev. 73 289–307.
  • Banks, D. L. (1996). A conversation with I. J. Good. Statist. Sci. 11 1–19.
  • Barnard, G. A. (1967). The Bayesian controversy in statistical inference. J. Inst. Actuar. 93 229–269.
  • Bayarri, M. J., Berger, J. O., Forte, A. and García-Donato, G. (2012). Criteria for Bayesian model choice with application to variable selection. Ann. Statist. 40 1550–1577.
  • Bennett, J. H. (1990). Statistical Inference and Analysis: Selected Correspondence of R. A. Fisher. The Clarendon Press, Oxford Univ. Press, New York.
  • Berger, J. O. (2006). Bayes factors. In Encyclopedia of Statistical Sciences, Vol. 1, 2nd ed. (S. Kotz, N. Balakrishnan, C. Read, B. Vidakovic and N. L. Johnson, eds.) 378–386. Wiley, Hoboken, NJ.
  • Broad, C. D. (1918). On the relation between induction and probability (Part I). Mind 27 389–404.
  • Clark, R. W. (1968). JBS: The Life and Work of J. B. S. Haldane. Coward-McCann, Inc., New York.
  • Crow, J. F. (1992). Centennial: JBS Haldane, 1892–1964. Genetics 130 1.
  • Cuthill, J. H. and Charleston, M. (in press). Wing patterning genes and coevolution of Müllerian mimicry in Heliconius butterflies: Support from phylogeography, co-phylogeny and divergence times. Evolution.
  • Dienes, Z. (2014). Using Bayes to get the most out of non-significant results. Front. Psychol. 5, 781.
  • Edwards, A. W. F. (1974). The history of likelihood. Int. Stat. Rev. 42 9–15.
  • Fienberg, S. E. (2006). When did Bayesian inference become “Bayesian”? Bayesian Anal. 1 1–40.
  • Fisher, R. A. (1932). Inverse probability and the use of likelihood. Math. Proc. Cambridge Philos. Soc. 28 257–261.
  • Fisher, R. A. (1934). Probability likelihood and quantity of information in the logic of uncertain inference. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 146 1–8.
  • Fouskakis, D., Ntzoufras, I. and Draper, D. (2015). Power-expected-posterior priors for variable selection in Gaussian linear models. Bayesian Anal. 10 75–107.
  • Good, I. J. (1958). Significance tests in parallel and in series. J. Amer. Statist. Assoc. 53 799–813.
  • Good, I. J. (1979). Studies in the history of probability and statistics. XXXVII. A. M. Turing’s statistical work in World War II. Biometrika 66 393–396.
  • Good, I. J. (1980). The contributions of Jeffreys to Bayesian statistics. In Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys (A. Zellner, ed.) 21–34. North-Holland, Amsterdam.
  • Good, I. J. (1988). The interface between statistics and philosophy of science. Statist. Sci. 3 386–412.
  • Haldane, J. B. S. (1919). The probable errors of calculated linkage values, and the most accurate method of determining gametic from certain zygotic series. J. Genet. 8 291–297.
  • Haldane, J. B. S. (1927). Possible Worlds: And Other Essays 52. Chatto & Windus.
  • Haldane, J. B. S. (1932). A note on inverse probability. Math. Proc. Cambridge Philos. Soc. 28 55–61.
  • Haldane, J. B. S. (1937). The exact value of the moments of the distribution of $\chi ^{2}$, used as a test of goodness of fit, when expectations are small. Biometrika 29 133–143.
  • Haldane, J. B. S. (1938). The approximate normalization of a class of frequency distributions. Biometrika 29 392–404.
  • Haldane, J. B. S. (1945). On a method of estimating frequencies. Biometrika 33 222–225.
  • Haldane, J. B. S. (1948). The precision of observed values of small frequencies. Biometrika 35 297–300.
  • Haldane, J. B. S. (1951). A class of efficient estimates of a parameter. Bull. Inst. Internat. Statist. 23 231–248.
  • Haldane, J. B. S. (1966). An autobiography in brief. Perspect. Biol. Med. 9 476–481.
  • Haldane, J. B. S. and Smith, C. A. B. (1947). A simple exact test for birth-order effect. Ann. Eugen. 14 117–124.
  • Hodgkin, D. C. and Jeffreys, H. (1976). Obituary. Nature 260 564.
  • Hoeting, J. A., Madigan, D., Raftery, A. E. and Volinsky, C. T. (1999). Bayesian model averaging: A tutorial. Statist. Sci. 14 382–417.
  • Holmes, C. C., Caron, F., Griffin, J. E. and Stephens, D. A. (2015). Two-sample Bayesian nonparametric hypothesis testing. Bayesian Anal. 10 297–320.
  • Howie, D. (2002). Interpreting Probability: Controversies and Developments in the Early Twentieth Century. Cambridge Univ. Press, Cambridge.
  • Jeffreys, H. (1931). Scientific Inference, 1st ed. Cambridge Univ. Press, Cambridge.
  • Jeffreys, H. (1933). On the prior probability in the theory of sampling. Math. Proc. Cambridge Philos. Soc. 29 83–87.
  • Jeffreys, H. (1935). Some tests of significance, treated by the theory of probability. Math. Proc. Cambridge Philos. Soc. 31 203–222.
  • Jeffreys, H. (1936a). Further significance tests. Math. Proc. Cambridge Philos. Soc. 32 416–445.
  • Jeffreys, H. (1936b). XXVIII. On some criticisms of the theory of probability. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 22 337–359.
  • Jeffreys, H. (1939). Theory of Probability. Oxford Univ. Press, Oxford.
  • Jeffreys, H. (1977). Probability theory in geophysics. IMA J. Appl. Math. 19 87–96.
  • Jeffreys, H. (1980). Some general points in probability theory. In Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys (A. Zellner, ed.) 451–453. North-Holland, Amsterdam.
  • Kass, R. E. (2009). Comment: The importance of Jeffreys’s legacy. Statist. Sci. 24 179–182.
  • Kass, R. E. and Raftery, A. E. (1995). Bayes factors. J. Amer. Statist. Assoc. 90 773–795.
  • Kendall, M. G., Bernoulli, D., Allen, C. G. and Euler, L. (1961). Studies in the history of probability and statistics: XI. Daniel Bernoulli on maximum likelihood. Biometrika 48.
  • Lai, D. C. (1998). John Burdon Sanderson Haldane (1892–1964) polymath beneath the firedamp: The story of JBS Haldane. Bull. Anesth. History 16 3–7.
  • Lambert, J. H. and DiLaura, D. L. (2001). Photometry, or, on the Measure and Gradations of Light, Colors, and Shade: Translation from the Latin of Photometria, Sive, de Mensura et Gradibus Luminis, Colorum et Umbrae. Illuminating Engineering Society of North America, New York.
  • Lane, D. A. (1980). Fisher, Jeffreys, and the nature of probability. In R. A. Fisher: An Appreciation. Lecture Notes in Statist. 1 148–160. Springer, New York.
  • Lattimer, J. M. and Steiner, A. W. (2014). Neutron star masses and radii from quiescent low-mass X-ray binaries. Astrophys. J. 784, 123.
  • Lindley, D. V. (1957). Binomial sampling schemes and the concept of information. Biometrika 44 179–186.
  • Lindley, D. (2009). Comment [MR2655841]. Statist. Sci. 24 183–184.
  • Ly, A., Verhagen, J. and Wagenmakers, E.-J. (2016). Harold Jeffreys’s default Bayes factor hypothesis tests: Explanation, extension, and application in psychology. J. Math. Psych. 72 19–32.
  • Malikov, E., Kumbhakar, S. C. and Tsionas, E. G. (2015). Bayesian approach to disentangling technical and environmental productivity. Econometrics 3 443–465.
  • Ostrogradski, M. V. (1848). On a problem concerning probabilities. St. Petersburg Acad. Sci. 6 321–346.
  • Pearson, K. (1892). The Grammar of Science 17. Walter Scott.
  • Peirce, C. S. (1878). The probability of induction. Pop. Sci. Mon. 12 705–718.
  • Pirie, N. W. (1966). John Burdon Sanderson Haldane. 1892–1964. Biogr. Mem. Fellows R. Soc. 12 219–249.
  • Prevost, P. and L\`Huilier, S. A. (1799). Sur les probabilités. Mémoires de L‘Academie Royale de Berlin 1796 117–142.
  • Robert, C. P. (2016). The expected demise of the Bayes factor. J. Math. Psych. 72 33–37.
  • Robert, C., Chopin, N. and Rousseau, J. (2009). Harold Jeffreys‘s theory of probability revisited. Statist. Sci. 24 141–172.
  • Sarkar, S. (1992). A centenary reassessment of JBS Haldane, 1892–1964. BioScience 42 777–785.
  • Senechal, M. (2012). I Died for Beauty: Dorothy Wrinch and the Cultures of Science. Oxford Univ. Press, Oxford.
  • Senn, S. (2009). Comment on “Harold Jeffreys’s theory of probability revisited”. Statist. Sci. 24 185–186.
  • Smith, J. M. (1992). JBS Haldane. In The Founders of Evolutionary Genetics 37–51. Springer, New York.
  • Sparks, D. K., Khare, K. and Ghosh, M. (2015). Necessary and sufficient conditions for high-dimensional posterior consistency under $g$-priors. Bayesian Anal. 10 627–664.
  • Taroni, F., Marquis, R., Schmittbuhl, M., Biedermann, A., Thiéry, A. and Bozza, S. (2014). Bayes factor for investigative assessment of selected handwriting features. Forensic Sci. Int. 242 266–273.
  • Terrot, C. (1853). Summation of a compound series, and its application to a problem in probabilities. Trans. R. Soc. Edinb. 20 541–545.
  • Todhunter, I. (1858). Algebra for the Use of Colleges and Schools: With Numerous Examples. Cambridge Univ. Press, Cambridge.
  • Todhunter, I. (1865). A History of the Mathematical Theory of Probability: From the Time of Pascal to That of Laplace/by I. Todhunter. Macmillan and Company.
  • Turing, A. M. (1941/2012). The Applications of Probability to Cryptography. UK National Archives, HW $25/37$.
  • Von Mises, R. (1931). Wahrscheinlichkeitsrechnung und Ihre Anwendung in der Statistik und Theorestischen Physik. Franz Deuticke.
  • White, M. J. D. (1965). JBS Haldane. Genetics 52 1.
  • Wrinch, D. and Jeffreys, H. (1919). On some aspects of the theory of probability. Philos. Mag. 38 715–731.
  • Wrinch, D. and Jeffreys, H. (1921). On certain fundamental principles of scientific inquiry. Philos. Mag. 42 369–390.
  • Wrinch, D. and Jeffreys, H. (1923a). On certain fundamental principles of scientific inquiry (second paper). Philos. Mag. 45 368–374.
  • Wrinch, D. and Jeffreys, H. (1923b). The theory of mensuration. Philos. Mag. 46.
  • Zabell, S. (1989). The rule of succession. Erkenntnis 31 283–321.
  • Zabell, S. (2012). Commentary on Alan M. Turing: The applications of probability to cryptography. Cryptologia 36 191–214.