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Some history of the hierarchical Bayesian methodology

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Trabajos de Estadistica Y de Investigacion Operativa

Summary

A standard technique in subjective “Bayesian” methodology is for a subject (“you”) to make judgements of the probabilities that a physical probability lies in various intervals. In the hierarchical Bayesian technique you make probability judgements (of a higher type, order, level, or stage) concerning the judgements of lower type. The paper will outlinesome of the history of this hierarchical technique with emphasis on the contributions by I. J. Good because I have read every word written by him.

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References

  • Bishop, Y.M.M., Fienberg, S.E., andHolland, P.W. (1975).Discrete Multivariate Analysis Harvard, Mass: M.I.T. Press.

    MATH  Google Scholar 

  • Crook, J.F. andGood, I.J. (1980). On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, II.Ann. Statist. (to be published).

  • David, F.N. (1949).Probability Theory for Statistical Methods. Cambridge: University Press.

    MATH  Google Scholar 

  • Dempster, A.P., Laird, N.M., andRubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm.J. Roy. Statist. Soc. B 39, 1–38 (with discussion).

    MATH  MathSciNet  Google Scholar 

  • Good, I.J. (1950).Probability and the Weighing of Evidence. London: Griffin.

    MATH  Google Scholar 

  • — (1952). Rational decisions.J. Roy Statist. Soc. B 14, 107–114

    MathSciNet  Google Scholar 

  • — (1953). On the population frequencies of species and the estimation of population parameters.Biometrika 40, 237–264.

    MATH  MathSciNet  Google Scholar 

  • — (1955). Contribution to the discussion on the Symposium on Linear Programming.J. Roy. Statist. Soc. B. 17, 194–196.

    MathSciNet  Google Scholar 

  • — (1956). On the estimation of small frequencies in contingency tables.J. Roy. Statist. Soc. B.,18, 113–124.

    MATH  MathSciNet  Google Scholar 

  • — (1957). Saddle-point methods for the multinomial distribution.Ann. Math. Statist. 28, 861–881.

    Article  MathSciNet  Google Scholar 

  • — (1959). Kinds of probability.Science 127, 443–447.

    Article  MathSciNet  Google Scholar 

  • Good, I.J., (1962). Subjective probability as the measure of a non-measurable set. InLogic, Methodology and Philosophy of Science (Nagel, E., Suppes, P., and Tarski, A. eds), 319–329.

  • — (1963). Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables.Ann. Math. Statist,34, 911–934.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1964). Contribution to the discussion of A.R. Thatcher Relationships between Bayesian and confidence limits for predictions.J. Roy. Statist. Soc. B,26, 204–205.

    Google Scholar 

  • — (1965).The Estimation of Probabilities: An Essay on Modern Bayesian Methods. Harvard, Mass: M.I.T. Press.

    MATH  Google Scholar 

  • — (1966). How to estimate probabilities.J. Inst. Math. Applics. 2, 364–383.

    Article  MATH  Google Scholar 

  • — (1967). A Bayesian significance test for multinomial distributions.J. Roy. Statist. Soc. B 29, 399–431.

    MATH  MathSciNet  Google Scholar 

  • — (1969). A subjective analysis of Bode’s law and an ‘objective’ test for approximate numerical rationality.J. Amer. Statist. Assoc. 64, 23–66 (with discussion).

    Article  Google Scholar 

  • Good, I.J., (1971a). Contribution to the discussion of Orear and Cassel (1971), 284–286.

  • — (1971b). Nonparametric roughness penalty for probability densities.Nature Physical Science 229, 29–30.

    Google Scholar 

  • — (1971c). Twenty-seven principles of rationality. InFoundations of Statistical Inference (V.P. Godambe and D.A. Sprott. ed.) 123–127, Toronto: Holt, Rinehart and Winston.

    Google Scholar 

  • — (1975). The Bayes factor against equiprobability of a multinomial population assuming a symmetric Dirichlet prior.Ann. Statist. 3, 246–250.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1976a). On the application of symmetric Dirichlet distributions and their mixtures to contingency tables.Ann. Statist. 4, 1159–1189.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1976b). The Bayesian influence or how to sweep subjectivism under the carpet. InFoundations of Probability Theory, Statistical Inference, and Statistical Theories of Science 2 (C.A. Hooker and W. Harper, eds.) 125–174, Dordrecht, Holland: D. Reidel.

    Google Scholar 

  • Good, I.J., (1979a). A comparison of some statistical estimates for the numbers of contingency tables, item C26 in “Comments, Conjectures, and Conclusions”. InJ. Statist. Comput. Simul. 8, 312–314.

    Google Scholar 

  • — (1979b). The contributions of Jeffreys to Bayesian statistics. InStudies in Bayesian Econometrics and Statistics in Honor of Harold Jeffreys. (A. Zellner, ed.), 21–34. Amsterdam: North Holland.

    Google Scholar 

  • — (1979c). Predictive sample reuse and the estimation of probabilities.J. Statist. Comput. Simul. 9, 238–239.

    Article  MathSciNet  Google Scholar 

  • Good, I.J. andCrook, J.F. (1974). The Bayes/non-Bayes compromise and the multinomial distribution.J. Amer. Statist. Assoc. 69, 711–720.

    Article  MATH  Google Scholar 

  • — (1977). The enumeration of arrays and a generalization related to contingency tables.Discrete Mathematics 19, 23–45.

    Article  MATH  MathSciNet  Google Scholar 

  • Good, I.J. andGaskins, R.A. (1971). Non-parametric roughness penalties for probability densities.Biometrika 58, 255–277.

    Article  MATH  MathSciNet  Google Scholar 

  • — (1972). Global nonparametric estimation of probability densities.Virginia J. of Science 23, 171–193.

    MathSciNet  Google Scholar 

  • — (1979). Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data.J. Amer. Statist. Assoc. 75, 42–73 (with discussion).

    Article  MathSciNet  Google Scholar 

  • Hardy, G.F. (1889). In correspondence in Insurance Record. Reprinted inTrans.Fac. Actuaries 8 (1920), 174–182.

    Google Scholar 

  • Hurwicz, L. (1951). Some specification problems and applications to econometric models,Econometrics 19, 343–344 (abstract).

    Google Scholar 

  • Jaynes, E.T. (1957). Information theory and statistical mechanics.Phys. Rev. 106, 620–630.

    Article  MathSciNet  Google Scholar 

  • Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems.Proc. Roy. Soc. (London), A. 186, 453–461.

    Article  MATH  MathSciNet  Google Scholar 

  • Johnson, W.E. (1932). Appendix (ed. R.B. Braithwaite) to Probability: deductive and inductive problems.Mind 41, 421–423.

    Google Scholar 

  • Kemble, E.C. (1941). The probability concept.Philosophy of Science 8, 204–232.

    Article  MathSciNet  Google Scholar 

  • Keynes, J.M. (1921).A Treatise on Probability. London: Macmillan.

    MATH  Google Scholar 

  • Koopmanm, B.O. (1940a). The basis of probability.Bull. Amer. Math. Soc. 46, 763–764.

    Article  MathSciNet  Google Scholar 

  • — (1940a). The axioms and algebra of intuitive probability.Ann. Math.,41, 269–292.

    Article  Google Scholar 

  • Leonard, T. (1978). Density estimation, stochastic processes and prior information.J. Roy. Statist. Soc, B,40, 113–146 (with discussion).

    MATH  MathSciNet  Google Scholar 

  • Levi, I. (1973). Inductive logic and the improvement of knowledge.Tech. Rep, Columbia University.

  • Levin, B. andReeds, J. (1977). Compound multinomial likelihood functions: proof of a conjecture of I.J. Good.,Ann. Statist. 5, 79–87.

    Article  MATH  MathSciNet  Google Scholar 

  • Lidstone, G.J. (1920). Note on the general case of the Bayes-Laplace formula for inductive or a posteriori probabilities.Trans. Fac. Actuar. 8, 182–192.

    Google Scholar 

  • Lindley, D.V. (1971). The estimation of many parameters. InFoundations of Statistical Inference (V.P. Godambe and D.A. Sprott, eds.) 435–455, (with discussion). Toronto: Holt, Rinehart and Winston.

    Google Scholar 

  • Lindley, D.V. andSmith, A.F.M. (1972). Bayes estimates for the linear model.J. Roy. Statist. Soc. B. 34, 1–41 (with discussion).

    MATH  MathSciNet  Google Scholar 

  • De Morgan, A. (1847). Theory of probabilities.Encyclopaedia Metropolitana 2, 393–490.

    Google Scholar 

  • Orear, J. andCassel, D. (1971). Applications of statistical inference to physics. InFoundations of Statistical Inference (V.P. Godambe and D.A. Sprott. eds.) 280–288 (with discussion). Toronto: Holt, Rinehart and Winston.

    Google Scholar 

  • Pelz, W. (1977).Topics on the estimation of small probabilities. Ph D thesis, Virginia Polytechnic Institute and State University.

  • Perks, W. (1947). Some observations on inverse probability including a new indifference rule.J. Inst. Actuar 73, 285–312.

    MathSciNet  Google Scholar 

  • Reichenbach, H. (1949).The Theory of Probability. Berkeley: University of California Press.

    MATH  Google Scholar 

  • Ribbins, H. (1951). Asymptotically subminimax solutions of compound statistical decision problems.Proc. 2nd Berkeley Symp. 131–148. Berkeley: University of California Press.

    Google Scholar 

  • — (1956). An empirical Bayes approach to statistics.Proc. 3rd Berkeley Symp.1, 157–163. Berkeley: University of California Press.

    Google Scholar 

  • Rogers, J.M. (1974).Some examples of compromises between Bayesian and non-Bayesian statistical methods. Ph. D. Thesis, Virginia Polytechnic Institute and State University.

  • Savage, L.J. (1954).The Foundations of Statistics. New York: Wiley.

    MATH  Google Scholar 

  • Smith, C.A.B. (1961). Consistency in statistical inference and decision.J. Roy. Statist. Soc. B. 23, 1–37 (with discussion).

    MATH  Google Scholar 

  • von Mises, R. (1942). On the correct use of Bayes’s formula.Ann. Math. Statist. 13, 156–165.

    Article  MATH  Google Scholar 

References in the Discussion

  • Dawes, R.M. (1971). A case study of graduate admissions: Application of three principles of human decision making.Amer. Psychol. 25, 180–188.

    Article  Google Scholar 

  • Garthwaithe, P. (1977).Psychological aspects of subjective probability elicitation. M.Sc. Thesis. Department of Statistics, University College of Wales, Aberystwyth.

    Google Scholar 

  • Geisser, S. (1975a). The predictive sample reuse method with applications.J. Amer. Statist. Assoc. 70, 320–328.

    Article  MATH  Google Scholar 

  • — (1975b). A new approach to the fundamental problem of applied statistics.Sankhya B 37, 385–397.

    MATH  MathSciNet  Google Scholar 

  • Goel, P.K. andDeGroot, M.H. (1979). Information about hyperparameters in hierarchical models.Tech. Rep. 160. Department of Statistics, Carnegie-Mellon University.

  • Gokhale andPress, S.J. (1979). The assessment of a prior distribution for the correlation coefficient in a bivariate normal distribution.Tech. Rep. 58, University of California, Riverside.

  • Good, I.J. (1962). A compromise between credibility and subjective probability.International Congress of Mathematicians, Abstracts of Short Communication. Stockholm, 160.

  • — (1965b). Speculations concerning the first ultraintelligent machine.Advances in Computer 6, 31–38.

    Google Scholar 

  • Good, I.J. (1971d). Unpublished lecture notes entitled “The Bayesian Influence” 122. Statistics Department, Virginia Polytechnic Institute and State University.

  • — (1972). Food for thought. InInterdisciplinary Investigation of the Brain (J.P. Nicholson, ed.) 1972, 213–228. New York: Plenum Press.

    Google Scholar 

  • — (1975b). Explicativity, corroboration and the relative odds of hypotheses.Synthese 30, 39–73.

    Article  MATH  Google Scholar 

  • — (1980a). The logic of hypothesis testing. InPhilosophical Foundations of Economics (J.C. Pitt, ed.) Dordrecht: Reidel.

    Google Scholar 

  • Good, I.J. (1980b). An approximation of value in the Bayesian analysis of contingency tables.J. Statist. Comput. Simulation (in press).

  • Kadane, J.B., Dickey, J.M., Winkler, R.L., Smith, W.S. andPeters, S.C. (1980). Interactive elicitation of opinion for a normal linear model.J. Amer. Statist. Assoc. 75 (to appear).

  • Leonard, T. (1973). A Bayesian method for histograms.Biometrika 60, 297–308.

    MATH  MathSciNet  Google Scholar 

  • — (1973). Bayesian estimation methods for two-way contingency tables.J. Roy. Statist. Soc. B 37, 23–37.

    MathSciNet  Google Scholar 

  • — (1977). A Bayesian approach to some multinomial estimation and pretesting problems.J. Amer. Statist. Assoc. 72, 869–874.

    Article  MATH  MathSciNet  Google Scholar 

  • Lindqvist, B. (1977). How fast does a Markov chain forget the initial state? A decision theoretical approach.Scand. J. Statist. 4, 145–152.

    MathSciNet  Google Scholar 

  • — (1978). On the loss of information incurred by lumping states of a Markov chain.Scand. J. Statist. 5, 92–98.

    MATH  MathSciNet  Google Scholar 

  • O’Hagan, A. andLeonard, T. (1976). Bayes estimation subject to uncertainty about parameter constraints.Biometrika 63, 201–203.

    Article  MATH  MathSciNet  Google Scholar 

  • Scott, D., Tapia, R.A. andThompson, J.R. (1978). Multivariate density estimation by discrete maximum penalized likelihood methods. InGraphical Representation of Multivariate Data. 169–182. New York: Academic Press.

    Google Scholar 

  • Slovic, P. andLichtenstein, S.C. (1971) Comparison of Bayesian and regression approaches to the study of information processing in judgement.Organizational Behavior and Human Performance 6, 649–744.

    Article  Google Scholar 

  • Stone, M. (1974). Cross-validation and multinomial prediction.Biometrika 61, 509–515.

    Article  MATH  MathSciNet  Google Scholar 

  • Tversky, A. (1974). Assessing uncertainty.J. Roy. Statist. Soc. B 36, 148–159.

    MATH  MathSciNet  Google Scholar 

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Good, I.J. Some history of the hierarchical Bayesian methodology. Trabajos de Estadistica Y de Investigacion Operativa 31, 489–519 (1980). https://doi.org/10.1007/BF02888365

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