Bayesian Analysis

Can a significance test be genuinely Bayesian?

Carlos A. de B. Pereira, Julio Michael Stern, and Sergio Wechsler

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The Full Bayesian Significance Test, FBST, is extensively reviewed. Its test statistic, a genuine Bayesian measure of evidence, is discussed in detail. Its behavior in some problems of statistical inference like testing for independence in contingency tables is discussed.

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Bayesian Anal., Volume 3, Number 1 (2008), 79-100.

First available in Project Euclid: 22 June 2012

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Zentralblatt MATH identifier

Bayes tests Sharp hypotheses Significance tests 62A15 62C10 62F03 62F15


Pereira, Carlos A. de B.; Stern, Julio Michael; Wechsler, Sergio. Can a significance test be genuinely Bayesian?. Bayesian Anal. 3 (2008), no. 1, 79--100. doi:10.1214/08-BA303.

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  • Basu, D. (1975). "Statistical Information and Likelihood (with discutions)." Sankhya A, 37: 1–71.
  • –- (1977). "On the elimination of nuisance parameters." Journal of the American Statistical Association, 72: 355–366.
  • Bernardo, J. and Smith, A. (1994). Bayesian Theory. NY: J Wiley.
  • Birnbaum, A. (1962). "On the foundations of statistical inference (with discussion)." Journal of the American Statistical Association, 57: 269–326.
  • Borges, W. and Stern, J. (2007). "The rules of logic composition for the Bayesian epistemic e-values." Logic J IGPL, 15: in press.
  • Cox, D. (1977). "The role of significance tests." Scandinavian J of Statistics, 4: 49–70.
  • Darwiche, A. and Ginsberg, M. (1992). "A Symbolic Generalization of Probability Theory." In AAAI92 Proc.. American Association for Articial Intelligence.
  • Dawid, A. and Lauritzen, S. (2001). "Compatible Prior Distributions." In Bayesian Methods with Applications to Science, Policy, and Official Statistics, EI George (editor), 109–118. Creta, Greece: ISBA.
  • DeGroot, M. (1975). Probability and Statistics. NY: Addison–Wesley, second edition.
  • Dempster, A. (1997). "The direct use of likelihood for significance testing." Statistics and Computing, 7: 247–252.
  • Fisher, R. (1922). "The mathematical foundations of theoretical statistics." Philosophical Transactions of the Royal Society, 222: 309–368.
  • –- (1934). Statistical Methods for Research Workers. Edinburgh, GB: Oliver and Boyd.
  • Good, I. (1983). Good Thinking: the Foundations of Probability and its Applications. Mineapolis, MN: University of Minnesota Press.
  • Goodman, L. and Kruskal, W. (1979). Measures of Association for cross classifications. New York: Springer.
  • Hwang, J., Casella, G., Wells, M., and Farrell, R. (1992). "Estimation of accuracy in testing." The Annals of Statistics, 20: 490–509.
  • Irony, T., Lauretto, M., Pereira, C., and Stern, J. (2002). "A Weibull wearout test: full Bayesian approach." In System Bayesian Reliability. Hayakawa, Irony, and Xie (Editors): In honor of Richard Barlow, 287–300. Singapure: World Scientific.
  • Irony, T. and Pereira, C. (1986). "Exact Bayes tests for equality of two proportions: Fisher versus Bayes." J Statistical Comptation and Simulation, 25: 93–114.
  • Irony, T., Pereira, C., and Tiwari, R. (2000). "Analysis of Opinion Swing: Comparison of Two Correlated Proportions." The American Statistician, 54(1): 57–62.
  • Jeffreys, H. (1939). Theory of Probability. Oxford: Claredon Press.
  • Kass, R. and Raftery, A. (1995). "Bayes Factors." Journal of the American Statistical Association, 90: 777–795.
  • Kempthorne, O. (1976). "Of what use are tests of significance and tests of hypothesis." Communications in Statistics - Theory and Methods, 8(A5): 763–777.
  • Kempthorne, O. and Folks, L. (1971). Probabilistic, Statistics, and Data Analysis. Ames, IO: The Iowa U Press.
  • Lauretto, M., Pereira, C., Stern, J., and Zacks, S. (2003). "Comparing parameters of two bivariate normal distributions using invariant Full Bayesian Significance Test." Brazilian J of Probability and Statistics, 17: 147–168.
  • Lindley, D. (1957). "A statistical paradox." Biometrika, 44(3): 187–192.
  • –- (1997). "Some comments on Bayes Factors." Journal of Statistical Planning and Inference, 61: 181–189.
  • Loschi, R., Monteiro, J., Rocha, G., and Mayrink, V. (2007). "Testing and estimating the non-conjunction fraction in meiosis I using reference priors." Biometrical Journal, 49(6): in press.
  • Madruga, M., Esteves, L., and Wechsler, S. (2001). "On the Bayesianity of Pereira-Stern tests." Test, 10: 291–299.
  • Madruga, M., Pereira, C., and Stern, J. (2003). "Bayesian Evidence Test for Precise Hypotheses." Journal of Statistical Planning and Inference, 117: 185–198.
  • Mcnemar, Q. (1947). "Note on the sampling error of the differences between correlated proportions or percentages." Psychometrika, 12: 153–157.
  • Montoya-Delgado, L., Irony, T., Pereira, C., and Whottle, M. (2001). "An unconditional exact test for the Hardy-Weimberg Equilibrium Law: Sample space ordering using the Bayes Factor." Genetics, 158: 875–883.
  • Neyman, J. and Pearson, E. (1936). "Sufficient statistics and uniformly most powerful tests of statistical hypotheses." Statistics Research Memoirs, 1: 133–137.
  • Pereira, C. and Lindley, D. (1987). "Examples questioning the use of partial likelihood." The Statistician, 36: 15–20.
  • Pereira, C., Nakano, F., Stern, J., and Whittle, M. (2006). "Genuine Bayesian multiallelic significance test for the Hardy-Weinberg equilibrium law." Genetics and Molecular Research, 5(4): 619–631.
  • Pereira, C. and Stern, J. (1999). "Evidence and credibility: full Bayesian significance test for precise hypotheses." Entropy, 1: 69–80.
  • –- (2001). "Full Bayesian Significance Tests for Coefficients of Variation." In Bayesian Methods with Applications to Science, Policy, and Official Statistics, EI George (editor), 391–400. Creta, Greece: ISBA.
  • –- (2001). "Model Selection: Full Bayesian Approach." Environmetrics, 12(6): 559–568.
  • Pereira, C. and Wechsler, S. (1993). "On the concept of P-value." Brazilian J Probability and Statistics, 7: 159–177.
  • Rodrigues, J. (2006). "Full Bayesian significance test for zero-inflated distributions." Communications in Statistics-Theory and Methods, 35: 1–9.
  • Rubin, H. (1987). "A weak system of axioms for rationality behavior and the non-separability of utility from prior." Statistical Decisions, 5: 47–58.
  • Schervish, M. (1995). Theory of Statistics. NY: Springer.
  • Shafer, G. (1982). "Lindley's paradox (with comments)." Journal of the American Statistical Association, 77: 325–351.
  • Stern, J. (2003). "Significance tests, Belief Calculi, and Burden of Proof in legal and Scientific Discourse." Frontiers in Artificial Intelligence and its Applications, 101: 139–147.
  • –- (2004). "Paraconsistent sensitive analysis for Bayesian significance tests." Lecture Notes in Artificial Intelligence, 3171: 134–143.
  • –- (2007). "Cognitive Constructivism, Eigen-Solutions, and Sharp Statistical Hypotheses." Cybernetics & Human Knowing, 14(1): 9–36.
  • Stern, J. and Zacks, S. (2002). "Testing the independence of Poisson variables under the Holgate bivariate distribution." Statistics and Probability Letters, 60: 313–320.
  • Wald, A. (1939). "Contributions to the theory of statistical estimation and testing hypotheses." Annals of Probability and Statistics, 10: 299–326.
  • –- (1950). Statistical Decision Functions. N York: Wiley.
  • Wechsler, S. (1993). "Exchangeability and Predictivism." Erkenntnis; International J Analytic Philosophy, 38(3): 343–350.
  • Wilks, S. (1935). "The likelihood test of independence in contingency tables." Annals of Mathematical Statistics, 6: 190–196.
  • –- (1938). "The large-sample distribution of the likelihood ratio for testing composite hypotheses." Annals of Mathematical Statistics, 9: 60–62.