The application of robust Bayesian analysis to hypothesis testing and Occam's Razor
Tóm tắt
Robust Bayesian analysis deals simultaneously with a class of possible prior distributions, instead of a single distribution. This paper concentrates on the surprising results that can be obtained when applying the theory to problems of testing precise hypotheses when the “objective” class of prior distributions is assumed. First, an example is given demonstrating the serious inadequacy of P-values for this problem. Next, it is shown how the approach can provide statistical quantification of Occam's Razor, the famous principle of science that advocates choice of the simpler of two hypothetical explanations of data. Finally, the theory is applied to multinomial testing.
Tài liệu tham khảo
Berger J. (1985),Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York.
Berger J., (1990), Robust Bayesian analysis: sensitivity to the prior.J. Statist. Planning and Inference, 25, 303–328.
Berger J. andDelampady, M., (1987), Testing precise hypotheses (with Discussion).Statistical Science, 2, 317–352.
Berger J. andSellke T. (1987), Testing of a point null hypothesis: the irreconciliability of significance levels and evidence (with Discussion).J. Amer. Statist. Assoc., 82, 112–139.
Brush S., (1989), Prediction and theory evaluation: The case of light bending.Science, 246, 1124–1129. See also the responses to this article inScience, 248, 422–423.
Casella G. andBerger R. L. (1987), Reconciling Bayesian and frequentist evidence in the one-sided testing problem.J. Amer. Statist. Assoc., 82, 106–111.
Degroot M. H., (1973), Doing what comes naturally: Interpreting a tail area as a posterior probability or as a like lihood ratio.J. Amer. Statist. Assoc., 68, 966–969.
Delampady M., (1989a), Lower bounds on Bayes factors for interval hypotheses.J. Amer. Statist. Assoc., 84, 120–124.
Delampady M. (1989b), Lower bounds on Bayes factors for invariant testing situations.J. Multivariate Anal. 28, 227–246.
Delampady M. andBerger J., (1990). Lower bounds on posterior probabilities for multinomial and chi-squared tests.Ann. Statist., 18, 1295–1316.
Dempster A. P., (1973), The direct use of likelihood for significance testing. InProc. of the Conference on Foundational Questions in Statistical Inference (O. Barndorff-Nielsen et al., eds.) 335–352. Dept. Theoretical Statistics, Univ. Aarhus.
Dickey J. M., (1977), Is the tail area useful as an approximate Bayes factor?J. Amer. Statist. Assoc. 72, 138–142.
Edwards W., Lindman H. andSavage L. J. (1963), Bayesian statistical inference for psychological research.Psychological Review, 70, 193–242.
Good I. J., (1967), A Bayesian significance test for the multinomial distribution.J. Roy. Statist. Soc. B, 29, 399–431.
Good I. J., (1983),Good Thinking: The Foundations of Probability and Its Applications. Univ. Minnesota Press, Minneapolis.
Good I. J., (1984), Notes C140, C144, C199, C200 and C201.J. Statist. Comput. Simulation, 19.
Gull S., (1988), Bayesian inductive inference and maximum entropy, In G. J. Erickson and C. R. Smith (eds.),Maximum Entropy and Bayesian Methods in Science and Engineering (Vol. 1) 53–74. Dordrecht: Kluwer Academic Publishers.
Jaynes E. T., (1979), Inference, method, and decision: Towards a Bayesian philosophy of science.J. Amer. Statist. Assoc., 74, 740–741.
Jefferys W. H., (1990), Bayesian analysis of random event generator data.Journal for Scientific Exploration, 4, 153–169.
Jeffreys H. (1921), Secular perturbations of the inner planets.Science, 54, 248.
Jeffreys H., (1939),Theory of Probability, Third Edition, (1961), Oxford: Clarendon Press.
Lindley D. V., (1957), A statistical paradox.Biometrika, 44, 187–192.
Loredo T. J., (1990), From Laplace to Supernova 1987 A: Bayesian inference in astrophysics. In P. Fougere (ed.),Maximum Entropy and Bayesian Methods, 81–142. Dordrecht: Kluwer Academic Publishers.
Nobili A. M. andWill C. M., (1986), The real value of Mercury's perihelion advance.Nature, 320, 39–41.
Poor C. L., (1921), The motions of the planets and the relativity theory.Science, 54, 30–34.
Roseveare N. T. (1982),Mercury's Perihelion from Le Verrier to Einstein, Oxford: Clarendon Press.
Shafer G. (1982), Lindley's paradox (with discussion).J. Amer. Statist. Assoc. 77, 325–351.
Smith, A. F. M. andSpiegelhalter D. J. (1980). Bayes factors and choice criteria for linear models.J. Roy. Statist. Soc. B, 42, 213–220.
Thorburn W. M., (1918). The myth of Occam's razor.Mind 27, 345–353.
Zellner A., (1984), Posterior odds ratios for regression hypotheses: General considerations and some specific results. InBasic Issues in Econometrics (A. Zellner, ed.), 275–305.