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Critical issues in different inferential paradigms

  • Ludovico Piccinato
Article

Summary

The main issues which characterize the current inferential paradigms are discussed. Emphasis is given to the kind of probability that can be used and to the problem of total or partial conditioning. Through classical examples, the major role of conditioning is stressed. Some trends of the main approaches (frequentist and Bayesian) are illustrated and some comments on the completely predictive approach are also provided.

Keywords

Bayesian inference Likelihood inference Frequentist inference Conditioning Principle of repeated sampling Likelihood Principle Decision theory Statistical models 

References

  1. Aitkin M. (1986), Statistical modelling: the likelihood approach.The Statistician 35, 103–13.CrossRefGoogle Scholar
  2. Alciati G., Di Bacco M., Drusini A. G. andPezzulli S. (1990), Determinazione dell'età alla morte mediante la frequenza degli osteoni secondari del femore,Quaderni di Anatomia Pratica, Serie XLVI, 85–110.Google Scholar
  3. Atkinson A. C. (1991), Optimum design of experiments, inStatistical Theory and Modelling in honour of Sir David Cox (eds. D. V. Hinkley, N. Reid, E. Snell) London: Chapman and Hall.Google Scholar
  4. Barlow R. E. andMendel M. B. (1992), Similarity as a characteristic of wear-out. To appear inReliability and Decision Making (eds. R. Barlow, C. A. Clarotti, F. Spizzichino), Barking (U.K.): Elsevier.Google Scholar
  5. Barnard G. A., Jenkins G. M. andWinstein C. B. (1962), Likelihood inference and time series.J.R. Statist. Soc. A, 125, 321–72.Google Scholar
  6. Bartholomew D. J. (1965), A comparison of some Bayesian and frequentist inferences.Biometrika 52, 19–35.zbMATHMathSciNetGoogle Scholar
  7. Basu D. (1975), Statistical information and likelihood.Sankhyà 37 A, 1–71.zbMATHGoogle Scholar
  8. Bayarri M. J., DeGroot M. H. andKadane J. B. (1988), What is the likelihood function? inStatistical Decision theory and Related Topics IV (eds. S. S. Gupta and J. O. Berger), 1, 3–16, New York: Springer-Verlag.Google Scholar
  9. Bayarri M. J. andDeGroot M. H. (1988), Discussion: Auxiliary parameters and simple likelihood functions, in Berger and Wolpert (1988)--om|-The Likelihood Principle, Hayward (Cal): Inst. Math. Statistics..Google Scholar
  10. Berger, J. O. (1985a).Statistical Decision Theory and Bayesian Analysis, New York: Springer-Verlag.zbMATHGoogle Scholar
  11. Berger, J. O. (1985b), The frequentist viewpoint and conditioning, inProc. Berkeley Conf. in honor of J. Neyman and J. Kiefer (eds. L. LeCam and R. A. Olshen), Belmont (Cal): Wadsworth.Google Scholar
  12. Berger J. O. (1990), Robust Bayesian analysis: sensitivity to the prior.J. Statist. Plann. and Inference 25, 303–28.zbMATHMathSciNetCrossRefGoogle Scholar
  13. Berger J. O. andBernardo J. M. (1992), On the development of reference priors inBayesian Statistics 4 (eds. J. M. Bernardo et al.), Oxford: Clarendon Press, 35–60.Google Scholar
  14. Berger J. O. andDelampady M. (1987), Testing precise hypotheses,Statist. Sci. 2, 317–35.zbMATHMathSciNetGoogle Scholar
  15. Berger J. O. andMortera J. (1991), Interpreting the stars in precise hypothesis testing.Int. Statist. Rev. 59, 337–53.zbMATHGoogle Scholar
  16. Berger J. O. andSellke T. (1987), Testing a point null hypothesis: The irreconcilability of P-values and evidence.J. Amer. Statist. Assoc. 82, 112–39.zbMATHMathSciNetCrossRefGoogle Scholar
  17. Berger, J. O. andWolpert R. L. (1988),The Likelihood Principle, Hayward (Cal): Inst. Math. Statistics.Google Scholar
  18. Berti P., Regazzini E. andRigo P. (1991), Coherent statistical inferences and the Bayes theorem,Ann. Statist. 19, 366–81.zbMATHMathSciNetGoogle Scholar
  19. Bertolino F., Piccinato L. andRacugno W. (1990), A marginal likelihood approach to analysis of variance,The Statistician 39, 415–24.CrossRefGoogle Scholar
  20. Birnbaum A (1962), On the foundations of statistical inference,J. Amer. Statist. Soc. 57, 269–326.MathSciNetGoogle Scholar
  21. Breslow N. (1990), Biostatistics and Bayes,Statistical Sci. 5, 269–98.zbMATHMathSciNetGoogle Scholar
  22. Brown L. D. (1990), An ancillarity paradox which appears in multiple linear regression,Ann. Statist. 18, 471–538.zbMATHMathSciNetGoogle Scholar
  23. Chaloner K. (1984), Optimal Bayesian experimental design for linear models.Ann. Statist. 12, 283–300.zbMATHMathSciNetGoogle Scholar
  24. Cifarelli D. M. andMuliere P. (1989),Statistica Bayesiana, Pavia: G. Iuculano.Google Scholar
  25. Cifarelli D. M. andRegazzini E. (1982), Some considerations about mathematical statistics teaching methodology suggested by the concept of exchangeability. InExchangeability in Probability and Statistics, (eds. G. Koch and F. Spizzichino), Amsterdam: North-Holland.Google Scholar
  26. Cifarelli D. M. andRegazzini E. (1987), Priors for exponential families which maximize the association between past and future observations, inProbability and Bayesian Statistics, (ed. R. Viertl), New York: Plenum Press.Google Scholar
  27. Clarotti C. A. andSpizzichino F. (1989), The Bayes predictive approach in reliability theory,IEEE Trans. on Reliability, 38, 379–82.zbMATHCrossRefGoogle Scholar
  28. Cochran W. G. (1963),Sampling techniques, New York: Wiley.Google Scholar
  29. Consonni G. andVeronese P. (1987), Coherent distributions and Lindley's paradox, inProbability and Bayesian Statistics, (ed. R. Viertl), New York: Plenum Press.Google Scholar
  30. Cox D. R. (1958), Some problems connected with statistical inference,Ann. Math. Statist. 9, 357–72.Google Scholar
  31. Cox D. R. (1975), Partial likelihood,Biometrika 62, 269–276.zbMATHMathSciNetCrossRefGoogle Scholar
  32. Cox D. R. (1978), Foundations of statistical inference: the case for electicism,Austrl. J. Statist. 20, 43–59.CrossRefGoogle Scholar
  33. Cox, D. R. andHinkley D. V. (1974),Theoretical Statistics, London: Chapman and Hall.zbMATHGoogle Scholar
  34. Daboni L. andWedlin A. (1982),Statistica. Un'introduzione alla impostazione neobayesiana, Torino: UTET.Google Scholar
  35. Dawid A. P. (1977), Invariant distributions and analysis of variance models,Biometrika 64, 291–7.zbMATHMathSciNetCrossRefGoogle Scholar
  36. Dawid A. P. (1983), Inference, Statistical I, inEnc. of Statistical Sciences (eds. S. Kotz, N. Johnson and C. B. Read) vol. 4, New York: Wiley.Google Scholar
  37. Dawid A. P. (1984), Statistical theory—the prequential approach.J. R. Statist. Soc. A 147, 278–92.zbMATHMathSciNetGoogle Scholar
  38. de Finetti B. (1937), La prévision: ses lois logiques, ses sources subjectives,Annales Inst. H. Poincaré, VII, 1–68 (English transl. eds. H. E. Kyburg and H. E. Smokler,Studies in Subjective Probability, New York: Wiley 1964).Google Scholar
  39. de Finetti B. (1937–8),Calcolo delle Probabilità, lecture notes (republished by Dip. di Matem. Appl. alle Sci. Econ. Statist. ed Attuariali “B. de Finetti”, Univ. di Trieste, 1986).Google Scholar
  40. de Finetti B. (1949), Sull'impostazione assiomatica del Calcolo delle Probabilità,Annali Triestini 19, sez. 2, 29–81 (English transl. in B. de Finetti:Probability, Induction and Statistics, New York: Wiley 1972).Google Scholar
  41. de Finetti B. (1970),Teoria delle probabilità, Torino: Einaudi (English transl.:Theory of Probability, New York: Wiley 1974–5).Google Scholar
  42. de Finetti B. (1971), Probabilità di una teoria e probabilità dei fatti, inStudi di Probabilità Statistica e Ricerca Operativa in onore di G. Pompilj, Ist. di Calcolo delle Probabilità, Univ. di Roma.Google Scholar
  43. DeGroot M. H. (1970),Optimal Statistical Decisions, New York: McGraw-Hill.zbMATHGoogle Scholar
  44. DeGroot M. H. (1982), Comments on the role of parameters in the predictive approach,Biometrics (supplement), 38, 86–91.CrossRefGoogle Scholar
  45. Di Bacco M. (1982), On the meaning of “true law” in statistical inference. InExchangeability in Probability and Statistics, (eds. G. Koch and F. Spizzichino), Amsterdam: North-Holland.Google Scholar
  46. Dickey J. (1973), Scientific reporting and personal probabilities: Student's hypothesis.J. Roy. Statist Soc. B 35, 285–305.MathSciNetGoogle Scholar
  47. DuMouchel, W. H. andHarris J. (1983), Bayes methods for combining the results of cancer studies in humans and other species.J. Amer. Statist. Assoc. 78, 293–315.zbMATHMathSciNetCrossRefGoogle Scholar
  48. Edwards A. W. F. (1972),Likelihood, Cambridge University Press.Google Scholar
  49. Efron B. (1982),The jackknife, the bootstrap, and other resampling plans, CBMSNSF Monographs, SIAM, Philadelphia (Penn).Google Scholar
  50. Efron B. (1986), Why isn't everyone a Bayesian?,Amer. Statistician 40, 1–11.zbMATHMathSciNetCrossRefGoogle Scholar
  51. Ferguson T. (1967),Mathematical Statistics. A Decision Theoretic Approach, New York: Academic Press.zbMATHGoogle Scholar
  52. Fisher R. A. (1922), On the mathematical foundations of theoretical statictics,Phil. Trans. R. Soc. of London A 22 (also inContributions to Mathematical Statitics, New York: Wiley, 1950, 309–68).Google Scholar
  53. Fisher R. A. (1925),Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd.Google Scholar
  54. Fisher R. A. (1955), Statistical methods and scientific induction,J. Roy. Statistics. Soc. B, 17, 69–78.zbMATHGoogle Scholar
  55. Ford, I., Titterington D. M. andKitsos C. (1989), Recent advances in non-linear experimental designs,Technometrics, 31, 49–60.zbMATHMathSciNetCrossRefGoogle Scholar
  56. Geisser S. (1980), A predictivistic primer, inBayesian Analysis in Econometrics and Statistics in honor of H. Jeffreys, (ed. A. Zellner), Amsterdam: North-Holland.Google Scholar
  57. Gelfand A. E. andSmith A. F. M. (1990), Sampling-based approaches to calculating marginal densities.J. Amer. Statist. Assoc. 85, 398–412.zbMATHMathSciNetCrossRefGoogle Scholar
  58. Gilio A. andScozzafava R. (1985), Vague, distributions in Bayesian testing of a null hypothesis,Metron 43, 167–74.zbMATHGoogle Scholar
  59. Good I. J. (1980), Some history of the hierarchical bayesian methodology, inBayesian Statistics (eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith), Valencia: University PressGoogle Scholar
  60. Hacking I. (1975),The Emergence of Probability, Cambridge Univ. Press.Google Scholar
  61. Hildreth C. (1963), Bayesian statisticians and remote clients,Econometrica 31, 422–438.CrossRefGoogle Scholar
  62. Hill, B. M. (1980), On finite additivity, non-conglomerability, and statistical paradoxes, inBayesian Statistics (eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith), Valencia: University PressGoogle Scholar
  63. Hill B. M. (1985–6), Some subjective Bayesian considerations in the selection of models,Econometric Reviews 4, 191–246.MathSciNetGoogle Scholar
  64. Hill B. M. (1987), The validity of the Likelihood Principle,Amer. Statistician 41, 95–100.zbMATHCrossRefGoogle Scholar
  65. Hill B. M. (1990), A theory of Bayesian data analysis, inBayesian and Likelihood Methods in Statistics and Econometrics, Essays in Honor of G. A. Barnard, (eds. di S. Geisser, J. S. Hodges, S. J. Press and A. Zellner), Amsterdam: North-HollandGoogle Scholar
  66. Hill B. M. (1992), Dutch books, the Jeffreys-Savage theory of hypothesis testing, and Bayesian reliability. To appear inReliability and Decision Making (eds., R. Barlow, C. A. Clarotti and F. Spizzichino), Barking (U.K.): Elsevier.Google Scholar
  67. Hinkley D. V. andReid, N. (1991), Statistical theory, inStatistical Theory and Modelling in honour of Sir David Cox, (eds. D. V. Hinkley, N. Reid and E. Snell), London: Chapman and Hall.Google Scholar
  68. Hodges J. S. (1987), Uncertainty, policy analysis and statistics,Statistical Sci. 2, 259–75.Google Scholar
  69. Huber P. J. (1981),Robust Statistics, New York: Wiley.zbMATHCrossRefGoogle Scholar
  70. Johnstone D. (1987), On the interpretation of hypothesis tests following Neyman and Pearson, inProbability and Bayesian Statistics, (ed. R. Viertl), New York: Plenum Press.Google Scholar
  71. Kalbfleisch J. D. (1975), Sufficiency and conditionality,Biometrika 62, 251–268.zbMATHMathSciNetCrossRefGoogle Scholar
  72. Kiefer J. (1975), General equivalence theorem for optimum designs (approximate theory),Ann. Statist. 2, 849–79.MathSciNetGoogle Scholar
  73. Kiefer J. (1977), Conditional confidence statements and confidence estimators,J. Amer. Statist. Assoc. 72, 789–811.zbMATHMathSciNetCrossRefGoogle Scholar
  74. Lane D. A. (1990), Conglomerability, coherence and countable additivity, inBayesian and Likelihood Methods in Statistics and Econometrics, Essays in Honor of G. A. Barnard, (eds. S. Geisser, J. S. Hodges, S. J. Press, and A. Zellner), Amsterdam: North-Holland.Google Scholar
  75. Leamer E. E. (1978),Specification Searches. Ad hoc Inferences with Nonexperimental Data, New York: Wiley.Google Scholar
  76. LeCam L. (1990), Maximum likelihood: an introduction,Int. Statist. Rev. 58, 153–171.CrossRefGoogle Scholar
  77. Lehmann E. (1983),Theory of Point Estimation New York: Wiley.zbMATHGoogle Scholar
  78. Lehmann E. (1986),Testing Statistical Hypotheses, New York: Wiley.zbMATHGoogle Scholar
  79. Lehmann E. (1988), Statistics: an overview, inEnc. of Statistical Sciences, (eds. S. Kotz, N. Johnson, C. B. Read), vol. 8, New York: Wiley.Google Scholar
  80. Lindley D. V. (1957), A statistical paradox,Biometrika 44, 187–92.zbMATHMathSciNetGoogle Scholar
  81. Lindley D. V. (1990), The present position in Bayesian statistics,Statistical Sci. 5, 44–89.zbMATHMathSciNetGoogle Scholar
  82. Lindley D. V., Smith A. F. M. (1972), Bayes estimates for the linear model,J. R. Statist. Soc. B 34, 1–41.zbMATHMathSciNetGoogle Scholar
  83. Liseo B. (1992), A note on a counterexample against the likelihood principle,Commun. Statist.-Theory and Methods 21 (2), 547–56.zbMATHMathSciNetGoogle Scholar
  84. Neyman J. (1956), Note on an article by sir Ronald Fisher,J. Roy. Statist. Soc. B 18, 288–94.zbMATHGoogle Scholar
  85. Neyman J. (1957), Inductive behavior as a basic concept of philosophy of science,Rev. Inst. Int. de Statistique 25, 7–22.CrossRefGoogle Scholar
  86. Pearson E. S. (1957), Statistical concepts in their relation to reality,J. Roy. Statist. Soc. B 17, 205–7.Google Scholar
  87. Pereira C. A. de B. andLindley D. V. (1987), Examples questioning the use of partial likelihood,The Statistician 36, 15–20.CrossRefGoogle Scholar
  88. Piccinato L. (1980) On the ordering of decision functions,Ist. Naz. di Alta Matemat.-Symposia Mathematica XXV London: Academic Press.Google Scholar
  89. Piccinato L. (1986), Finetti's logic of uncertainty and its impact on statistical thinking and practice, inBayesian Inference and Decision Techniques, Essays in honor of B. de Finetti (eds. P. K. Goel and A. Zellner), Amsterdam: North-Holland.Google Scholar
  90. Piccinato L. (1990), Sull'interpretazione del livello di significatività osservato, inScritti in omaggio a L. Daboni, Trieste: LINT.Google Scholar
  91. Racine A., Grieve A. P., Flühler H. andSmith A. F. M. (1986), Bayesian methods in practice: experiences in the pharmaceutical industry,Applied Statistics 35, 93–150.zbMATHCrossRefGoogle Scholar
  92. Raiffa H. andSchlaifer R. (1961),Applied Statistica Decision Theory, Cambridge (Mass): MIT Press.Google Scholar
  93. Rao M. M. (1988), Paradoxes in conditional probability,J. of Multivariate Analysis 27, 434–46.zbMATHCrossRefGoogle Scholar
  94. Regazzini E. (1983),Sulle probabilità coerenti nel senso di de Finetti, Bologna: CLUEB.Google Scholar
  95. Regazzini E. (1987), de Finetti's coherence and statistical inference,Annals of Statistics 15, 845–64.zbMATHMathSciNetGoogle Scholar
  96. Regazzini E. (1990), Qualche osservazione sulla scambiabilità e tecniche bayesiane,Atti XXXV Riunione Sci. della SIS, vol. 2, Padova: Cedam.Google Scholar
  97. Roberts H. V. (1974), Reporting of Bayesian studies, inStudies in Bayesian Econometrics and Statistics (eds. S. E. Fienberg and A. Zellner), Amsterdam: North-Holland, 465–83.Google Scholar
  98. Savage L. J. (1962), Subjective probability and statistical practice, inThe Foundations of Statistical Inference, (eds. L. J. Savage and others), London: Methuen.Google Scholar
  99. Savage L. J. (1976), On rereading R. A. Fisher,Ann. Statist. 4, 441–500.zbMATHMathSciNetGoogle Scholar
  100. Scardovi I. andMonari P. (1984), Statical induction: probable knowledge or optimal strategy?,Epistemologia 7, 101–120.Google Scholar
  101. Scozzafava R. (1981), Un esempio concreto di probabilità non σ-additiva: la distribuzione della prima cifra significativa dei dati statistici.Boll. Un. Matem. Ital. 18-A, 403–10.MathSciNetGoogle Scholar
  102. Scozzafava R. (1982), Probabilità σ-additive e non,Boll. Un. Matem. Ital. I-A 1–33.MathSciNetGoogle Scholar
  103. Scozzafava R. (1988), Probabilità qualitativa: teoria ed applicazioni della pseudodensità,Rend. Sem. Matem. e Fisico di Milano, 58, 101–14.zbMATHMathSciNetGoogle Scholar
  104. Scozzafava R. (1990). Probabilità condizionate: de Finetti o Kolmogorov?, inSritti in omaggio a L. Daboni, Trieste: LINT.Google Scholar
  105. Shafer G. (1982), Lindley's paradox,J. Amer. Statist. Assoc. 77, 325–51.zbMATHMathSciNetCrossRefGoogle Scholar
  106. Smith A. F. M. (1973), A general Bayesian linear model.J. R. Statist. Soc. B 35, 67–75.zbMATHGoogle Scholar
  107. Smith A. F. M. (1984), Bayesian statistics,J. R. Statist. Soc. A 147, 245–59.zbMATHCrossRefGoogle Scholar
  108. Smith A. F. M., Skene A., Shaw J. E. H. andNaylor J. C. (1987), Progress with numerical and graphical methods for practical Bayesian statistics,The Statistician 36, 75–82.CrossRefGoogle Scholar
  109. Speigelhalter D. J. (1987), Probabilistic expert system in medicine: practical issue in handling uncertainty,Statistical Sci. 2, 3–15.Google Scholar
  110. Verdinelli I. (1992), Advances in Bayesian experimental design. InBayesian Statistics 4 (eds. J. M. Bernardo et al.), Oxford: Clarendon Press, 467–82.Google Scholar
  111. Wald A. (1950),Statistical Decision Functions, New York: Wiley.zbMATHGoogle Scholar
  112. Wasserman L. (1992), Recent methodological advances in robust Bayesian inference. InBayesian Statistics 4 (eds. J. M. Bernardo et al.), Oxford: Clarendon Press, 483–502.Google Scholar

Copyright information

© Societa Italiana di Statistica 1992

Authors and Affiliations

  • Ludovico Piccinato
    • 1
  1. 1.Dipartimento di Statistica Probab. e Stat. ApplicateUniversità «La Sapienza»RomaItaly

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