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Bayesian Nonparametric Prediction and Statistical Inference

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Bayesian Analysis in Statistics and Econometrics

Part of the book series: Lecture Notes in Statistics ((LNS,volume 75))

Abstract

The problem of Bayesian nonparametric prediction and statistical inference is formulated and discussed. A solution is proposed based upon A n and H n as in Hill (1968, 1988). This solution gives rise to the posterior distribution of the number of species (or more generally, the number of distinct values) in a finite population of Hill (1968, 1979), and to the posterior distribution of percentiles of the population of Hill (1968). Next, the meaning of parameters in the subjective Bayesian theory of Bruno de Finetti is discussed in connection both with H n and with conventional parametric models. It is argued that the usual sharp distinction between prediction and parametric inference is largely illusory. The finite version of de Finetti’s theorem is emphasized for the practice of statistics, with the infinite case used only to obtain approximations and insight.

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References

  • Aitchison, J., and Dunsmore, I. R. (1975), Statistical Prediction Analysis, Cambridge University Press.

    Google Scholar 

  • Berger, J. (1984), The robust Bayesian viewpoint,” in Robustness of Bayesian Analysis, J. Kadane, ed., North-Holland: Amsterdam, (with discussion), 321–372.

    Google Scholar 

  • Berkson, J. (1938), Some difficulties of interpretation encountered in the application of the chi-square test.” Journal of the American Statistical Association, 33, 526–542.

    Article  MATH  Google Scholar 

  • Berliner, L. M., and Hill, B. M. (1988), Bayesian nonparametric survival analysis, Journal of the American Statistical Association, 83, 772–784 (with discussion).

    Article  MathSciNet  MATH  Google Scholar 

  • Blackwell, D., and MacQueen, J. B. (1973), Ferguson distributions via Pólya urn schemes, The Annals of Statistics, 1, 353–355.

    Article  MathSciNet  MATH  Google Scholar 

  • Boender, C. G. E., and Kan, A. H. G. Rinnooy (1987), A multinomial Bayesian approach to the estimation of population and vocabulary size, Biometrika, 74, 849–856.

    Article  MathSciNet  MATH  Google Scholar 

  • Borel, E. (1906), Sur les principes de le théorie cinétique des gaz, Annales de lÉcole Normale Supérieure, 9–32. Reprinted in Selecta: Jubilé Scient de M. Émile Borei, Gauthier-Villars, Paris, 1940, 243–265.

    Google Scholar 

  • Borel, E. (1914). Introduction Géométrique à Quelques Théories Physiques, Gauthier-Villars, Paris.

    Google Scholar 

  • Box, G. E. P., and Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, Addison-Wesley, Reading.

    MATH  Google Scholar 

  • Chang, C. (1988), Bayesian Nonparametric Prediction Based on Censored Data, University of Michigan Doctoral Dissertation.

    Google Scholar 

  • Chen, Wen-Chen (1978), On Zipf’s Law, University of Michigan Doctoral Dissertation.

    Google Scholar 

  • Chen, Wen-Chen (1980), On the weak form of Zipf’s law, Journal of Applied Probability, 17, 611–622.

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, Wen-Chen, Hill, B. M., Greenhouse, J., and Fayos, J. (1985). Bayesian analysis of survival curves for cancer patients following treatment (with discussion), in Bayesian Statistics 2, J. M. Bernardo, M. H. DeGroot, D. V. Lindley, A. F. M. Smith eds. North-Holland

    Google Scholar 

  • de Finetti, B. (1937), La prévision: ses lois logiques, ses sources subjectives, Annales de l’Institut Henri Poincaré, 7, 1–68.

    MATH  Google Scholar 

  • de Finetti, B. (1974). Theory of Probability, Vol 1, London: John Wiley & Sons, Inc.

    MATH  Google Scholar 

  • Dempster, A. P. (1963), On Direct Probabilities, Journal of the Royal Statistical Society B, 25, 100–114.

    MathSciNet  MATH  Google Scholar 

  • Diaconis, P., and Freedman, D. (1980), Finite exchangeable sequences, The Annals of Probability, 8, 745–764.

    Article  MathSciNet  MATH  Google Scholar 

  • Diaconis, P., and Freedman, D. (1981), Partial exchangeability and sufficiency, Proceedings of the Indian Statistical Institute Golden Jubilee International Conference on Statistics: Applications and New Directions, 205–236.

    Google Scholar 

  • Dickey, J., and Kadane, J. (1980), Bayesian decision theory and the simplification of models, in Evaluation of Econometric Models, J. Kmenta and J. Ramsey, eds., Academic Press, 245–268.

    Google Scholar 

  • Einstein, A. (1921). Geometrie und Erfahrung. J. Springer, Berlin.

    MATH  Google Scholar 

  • Feller, W. (1968), An Introduction to Probability Theory and its Applications, Third Edition, Revised Printing, New York: John Wiley & Sons.

    Google Scholar 

  • Feller, W. (1971), An Introduction to Probability Theory and its Applications, Vol. 2, Second Edition, New York: John Wiley & Sons.

    MATH  Google Scholar 

  • Ferguson, T. (1973), A Bayesian analysis of some nonparametric problems, The Annals of Statistics, 1, 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher, R. A. (1939), Student, Annals of Eugenics, 9, 1–9.

    Article  Google Scholar 

  • Fisher, R. A. (1948), Conclusions Fiduciare, Annales de l’Institut Henri Poincaré, 10, 191–213.

    Google Scholar 

  • Fisher, R. A. (1959). Statistical Methods and Scientific Inference, Second Edition, New York: Hafner Publishing Co.

    Google Scholar 

  • Good, I. J. (1965), The Estimation of Probabilities, MIT Research Monograph No. 30.

    Google Scholar 

  • Hacking, I. (1967), Slightly more realistic personal probability, Philosophy of Science, 34, 311–325.

    Article  Google Scholar 

  • Hartigan, J. (1983), Bayes Theory, New York: Springer-Verlag.

    MATH  Google Scholar 

  • Heath, D., and Sudderth, W. (1976), de Finetti’s theorem for exchangeable random variables, The American Statistician, 30, 188–189.

    Article  MathSciNet  MATH  Google Scholar 

  • Hewitt, E., and Savage, L. J. (1955), Symmetric measures on cartesian products, in The Writings of Leonard Jimmie Savage-A Memorial Selection, Published by The American Statistical Association and The Institute of Mathematical Statistics, 1981, 244–275.

    Google Scholar 

  • Hill, B. M. (1965), Inference about variance components in the one-way model, Jour. Amer. Statist. Assoc., 58, 918–932.

    Article  Google Scholar 

  • Hill, B. M. (1967), Correlated errors in the random model, Journal of the Americann Statistical Association, 62, 1387–1400.

    Article  MATH  Google Scholar 

  • Hill, B. M. (1968), Posterior distribution of percentiles: Bayes theorem for sampling from a finite population, Journal of the American Statistical Association, 63, 677–691.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1969), Foundations for the theory of least squares, Journal of the Royal Statistical Society, Series B, 31, 89–97.

    MATH  Google Scholar 

  • Hill, B. M. (1970), Zipf’s law and prior distributions for the composition of a population, Journal of the American Statistical Association, 65, 1220–1232.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1974a), The rank frequency form of Zipf’s law, Journal of the American Statistical Association, 69, 1017–1026.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1974b), On coherence, inadmissibility and inference about many parameters in the theory of least squares, in Studies in Bayesian Econometrics and Statistics in Honor of L. J. Savage, S. Fienberg and A. Zellner, eds., North Holland, 555–584.

    Google Scholar 

  • Hill, B. M. (1975a), A simple general approach to inference about the tail of a distribution, Annals of Statistics, 3, 1163–1174.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1975b), Aberrant behavior of the likelihood function in discrete cases, Journal of the American Statistical Association, 70, 717–719.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1977), Exact and approximate Bayesian solutions for inference about variance components and multivariate inadmissibility, in New Developments in the Application of Bayesian Methods, A. Aykac and C. Brumat, eds., North Holland, 129–152.

    Google Scholar 

  • Hill, B. M., (1978), Decision theory. In Studies in Statistics, Vol. 19, R.V. Hogg, ed., The Mathematical Association of America, 168–209.

    Google Scholar 

  • Hill, B. M. (1979), Posterior moments of the number of species in a finite population, and the posterior probability of finding a new species, Journal of the American Statistical Association, 74, 668–673.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1980a), Invariance and robustness of the posterior distribution of characteristics of a finite population, with reference to contingency tables and the sampling of species. In Bayesian Analysis in Econometrics and Statistics: Essays in Honor of Harold Jeffreys, ed A. Zellner, North-Holland, 383–395.

    Google Scholar 

  • Hill, B. M. (1980b), Robust analysis of the random model and weighted least squares regression, in Evaluation of Econometric Models, ed. by J. Kmenta and J. Ramsey, Academic Press, 197–217.

    Google Scholar 

  • Hill, B. M. (1980c), On finite additivity, non-conglomerability, and statistical paradoxes, (with discussion) in Bayesian Statistics, J. M. Bernardo, M. H. Degroot, D. V. Lindley, A. F. M. Smith, eds., University Press: Valencia, Spain, 39–66.

    Google Scholar 

  • Hill, B. M. (1981), A theoretical development of the Zipf (Pareto) law, in Studies on Zipf’s Laws, ed. by H. Guiter, Sprachwissenschaftliches Institute, Ruhr-Universitat, Bochun.

    Google Scholar 

  • Hill, B. M. (1985–86), Some subjective Bayesian considerations in the selection of models, Econometric Reviews 4, No. 2, 191–288 (with discussion).

    Article  MathSciNet  Google Scholar 

  • Hill, B. M. (1987a), The validity of the likelihood principle, The American Statistician, 41, 95–100.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1987b), Parametric models for A n: Splitting Processes and Mixtures, Unpublished, Department of Statistics, The University of Michigan.

    Google Scholar 

  • Hill, B. M. (1988), De Finetti’s theorem, induction, and A n, or Bayesian nonparametric predictive inference, in Bayesian Statistics 3, J. M. Bernardo, M. H. Degroot, D. V. Lindley, and A. F. M. Smith, eds., Oxford University Press, 211–241 (with discussion).

    Google Scholar 

  • Hill, B. M., (1990a), A theory of Bayesian data analysis, in Bayesian and Likelihood Methods in Statistics and Econometrics: Essays in Honor of George A. Barnard, S. Geisser, J. Hodges, S. J. Press, A. Zellner (Eds.), North-Holland, 1990, 49–73.

    Google Scholar 

  • Hill, B. M., (1990b), Bayesian Statistics, in 1990 Yearbook of Encyclopedia of Physical Science and Technology, Academic Press.

    Google Scholar 

  • Hill, B. M. (1990c), Discussion of An ancillarity paradox that appears in multiple linear regression, by L. D. Brown, in The Annals of Statistics, 1990.

    Google Scholar 

  • Hill, B. M. (1990d), Dutch books, the Jeffreys-Savage theory of hypothesis testing, and Bayesian reliability. Unpublished technical report, the University of Michigan. Presented at the course-congress on Reliability and Decision Making, Siena, Italy.

    Google Scholar 

  • Hill, B. M. (1990e), Robustness in Statistics. Unpublished technical report, the University of Michigan.

    Google Scholar 

  • Hill, B. M. and Lane, David (1985), Conglomerability and countable additivity, Sankhyá, 47, Series A, 366–379.

    MathSciNet  MATH  Google Scholar 

  • Hill, B. M., Lane, David, and Sudderth, William (1980), A strong law for some generalized urn processes, Annals of Probability, 8, 214–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M., Lane, David, and Sudderth, William (1987), Exchangeable urn processes, Annals of Probability, 15, 1586–1592.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. and Woodroofe, M. (1975), Stronger forms of Zipf’s law, Journ al of the American Statistical Association, 70, 212–219.

    Article  MathSciNet  MATH  Google Scholar 

  • Hoppe, F. (1987), The sampling theory of neutral alleles and an urn model in population genetics, Jour. of Mathematical Biology, 25, 123–159.

    Article  MathSciNet  MATH  Google Scholar 

  • Hume, David (1748), An Enquiry Concerning Human Understanding, London.

    Google Scholar 

  • Ijiri, Y., and Simon, H. A. (1975), Some distributions associated with Bose-Einstein statistics, Proceeding of the National Academy of Science, USA, 72, 1654–1657.

    MathSciNet  Google Scholar 

  • Jeffreys, H. (1957), Scientific Inference, Second Edition, Cambridge University Press.

    Google Scholar 

  • Jeffreys, H. (1961), Theory of Probability, Third Edition, Oxford at the Clarendon Press.

    Google Scholar 

  • Johnson, W. E. (1932), Probability: the deductive and inductive problems, Mind, 49, 409–423. [Appendix on pages 421–423 edited by R. B. Braithwaite].

    Article  Google Scholar 

  • Kingman, J. F. C. (1975), Random discrete distributions, Journal of the Royal Statistical Society, Series B, 37, 1–22 (with discussion).

    MathSciNet  MATH  Google Scholar 

  • Lane, D., and Sudderth, W. (1978), Diffuse models for sampling and predictive inference, Annals of Statistics, 6, 1318–1336.

    Article  MathSciNet  MATH  Google Scholar 

  • Lane, D., and Sudderth, W. (1984), Coherent predictive inference, Sankhyá Ser. A, 46, 166–185.

    MathSciNet  MATH  Google Scholar 

  • Lenk, P. (1984), Bayesian Nonparametric Predictive Distributions, Doctoral Dissertation, The University of Michigan.

    Google Scholar 

  • Lewins, W. A., and Joanes, D. N. (1984), Bayesian estimation of the number of species, Biometrics, 40, 323–328.

    Article  Google Scholar 

  • Lindley, D., and Smith, A. F. M. (1972), Bayes estimates for the linear model, Journal of the Royal Statistical Society, Series B, 34, 1–41.

    MathSciNet  MATH  Google Scholar 

  • Luce, R. D., Narens, L. (1987), Measurement scales on the continuum, Science, 236, 1527–1531.

    Article  MathSciNet  Google Scholar 

  • Mandelbrot, B. B. (1982), The Fractal Geometry of Nature, W. H. Freeman and Co., New York.

    MATH  Google Scholar 

  • Poincaré, H. (1912), Calcul des Probabilités, Deuxiéme Edition, Gauthier-Villars.

    Google Scholar 

  • Ramakrishnan, S. and Sudderth, W. (1988), A sequence of coin-toss variables for which the strong law fails, American Mathematical Monthly, 95, 939–941.

    Article  MathSciNet  MATH  Google Scholar 

  • Rényi, A. (1970), Probability Theory, New York: American Elsevier.

    Google Scholar 

  • Savage, L. J., 1961. The Subjective Basis of Statistical Practice. Unpublished technical report, Department of Mathematics, University of Michigan.

    Google Scholar 

  • Savage, L. J. et al., 1962. The Foundations of Statistical Inference. Methuen, London.

    Google Scholar 

  • Savage, L. J. (1972), The Foundations of Statistics, Second Revised Edition, New York: Dover.

    MATH  Google Scholar 

  • Schervish, M., Seidenfeld, T., and Kadane, J., (1984), The extent of non-conglomerability, Z. f. Wahrscheinlichkeitstheorie, 66, 205–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, A. F. M., (1986), Some Bayesian thoughts on modelling and model choice, The Statistician, 35, 97–102.

    Article  Google Scholar 

  • Whitrow, G. J. (1980), The Natural Philosophy of Time, Second Edition, Oxford University Press.

    Google Scholar 

  • Woodroofe, M. and Hill, B. M. (1975), On Zipf’s law, Journal of Applied Probability, 12, 425–434.

    Article  MathSciNet  MATH  Google Scholar 

  • Zabell, S. L. (1982), W. E. Johnson’s sufficientness postulate, The Annals of Statistics, 10, 1091–1099.

    Article  MathSciNet  MATH  Google Scholar 

Additional References

  • Berger, J. (1985), Statistical Decision Theory and Bayesian Analysis, Second Edition, Springer-Verlag.

    Google Scholar 

  • Berger, J. and Wolpert, R. (1988), The Likelihood Principle, Second Edition, IMS Lecture Notes-Monograph Series.

    Google Scholar 

  • de Finetti, B. (1975) Theory of Probability, Vol. 1, London: John Wiley & Sons, Inc.

    MATH  Google Scholar 

  • Heath, D., and Sudderth, W. (1978), “On finitely additive priors, coherence, and extended admissibility,” Annals Statist., 6, 333–345.

    Article  MathSciNet  MATH  Google Scholar 

  • Hill, B. M. (1980d), Review of Specification Searches, by Edward Learner, Journal of the American Statistical Association, 75, 252–253.

    Google Scholar 

  • Kolmogorov, A. N. (1950), Foundations of Probability, New York: Chelsea Publishing Co.

    Google Scholar 

  • Lindley, D. (1971), Bayesian Statistics, A Review, Philadelphia: SIAM.

    MATH  Google Scholar 

  • Robinson, G. K. (1979). Conditional properties of statistical procedures. Ann. Statist 7, 742–755.

    Article  MathSciNet  MATH  Google Scholar 

  • Samuels, S. (1989), “Comment: Who will solve the secretary problem,” Statistical Science, 4, 289–229

    Article  MathSciNet  Google Scholar 

  • Scozzafava, R. (1984), “A survey of some common misunderstandings concerning the role and meaning of finitely additive probabilities in statistical inference,” Statistica, anno XLIV, 21–45.

    Google Scholar 

  • Zabell, S. (1988), Comment on “De Finetti’s theorem, induction, and A n, or Bayesian nonparametric predictive inference.” In Bayesian Statistics 3, J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, eds., Oxford University Press, 233–236.

    Google Scholar 

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Authors and Affiliations

  1. University of Michigan, USA

    Bruce M. Hill

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  1. Bruce M. Hill
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  1. Department of Statistics, The Ohio State University, Columbus, OH, 43210-1247, USA

    Prem K. Goel

  2. Economic Analysis Unit, Indian Statistical Institute, Bangalore, 8 Mile Mysole Road, R.V. College — P.O., Bangalore, 560059, India

    N. Sreenivas Iyengar

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Hill, B.M. (1992). Bayesian Nonparametric Prediction and Statistical Inference. In: Goel, P.K., Iyengar, N.S. (eds) Bayesian Analysis in Statistics and Econometrics. Lecture Notes in Statistics, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2944-5_4

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  • DOI: https://doi.org/10.1007/978-1-4612-2944-5_4

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