Kociecki, Andrzej (2011): Some Remarks on Consistency and Strong Inconsistency of Bayesian Inference.

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Abstract
The paper provides new sufficient conditions for consistent and coherent Bayesian inference when a model is invariant under some group of transformations. Building on our theoretical results we reexamine an example from Stone (1976) giving some new insights. The priors for multivariate normal models and Structural Vector AutoRegression models that entail consistent and coherent Bayesian inference are also discussed.
Item Type:  MPRA Paper 

Original Title:  Some Remarks on Consistency and Strong Inconsistency of Bayesian Inference 
Language:  English 
Keywords:  invariant models; coherence; strong inconsistency; groups 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C11  Bayesian Analysis: General 
Item ID:  28731 
Depositing User:  Andrzej Kociecki 
Date Deposited:  13. Feb 2011 19:14 
Last Modified:  21. Feb 2013 05:07 
References:  Anderson, T.W. (2003), An Introduction to Multivariate Statistical Analysis, Third edition, John Wiley & Sons, New Jersey. Bondar, J.V. (1976), “Borel Cross–Sections and Maximal Invariants”, The Annals of Statistics, 4, pp. 866–877. Bondar, J.V. (1977), “A Conditional Confidence Principle”, The Annals of Statistics, 5, pp. 881–891. Bondar, J.V. and P. Milnes (1981), “Amenability: A Survey for Statistical Applications of Hunt–Stein and Related Conditions on Groups”, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , 57, pp. 103–128. Chang, T. and D. Eaves (1990), “Reference Priors for the Orbit in a Group Model”, The Annals of Statistics, 18, pp. 1595–1614. Dawid, A.P., M. Stone and J.V. Zidek (1973), “Marginalization Paradoxes in Bayesian and Structural Inference”, Journal of the Royal Statistical Society, series B, 35, pp. 189–213. Drèze, J.H. (1976), “Bayesian Limited Information Analysis of the Simultaneous Equations Model”, Econometrica, 44, pp. 1045–1075. Eaton, M.L. (1989), Group Invariance Applications in Statistics, Regional Conference Series in Probability and Statistics, vol. 1., Institute of Mathematical Statistics, Hayward, California. Eaton, M.L. (2008), “Dutch Book in Simple Multivariate Normal Prediction: Another Look”, in: D. Nolan and T. Speed, eds., Probability and Statistics: Essays in Honor of David A. Freedman, Institute of Mathematical Statistics, Beachwood, Ohio. Eaton, M.L. and D.A. Freedman (2004), “Dutch Book Against Some ‘Objective’ Priors”, Bernoulli, 10, pp. 861–872. Eaton, M.L. and W.D. Sudderth (1993), “Prediction in a Multivariate Normal Setting: Coherence and Incoherence, Sankhya, series A, 55, pp. 481–493. Eaton, M.L. and W.D. Sudderth (1995), “The Formal Posterior of a Standard Flat Prior in MANOVA is Incoherent”, Journal of the Italian Statistical Society, 2, pp. 251–270. Eaton, M.L. and W.D. Sudderth (1998), “A New Predictive Distribution for Normal Multivariate Linear Models”, Sankhya, series A, 60, pp. 363–382. Eaton, M.L. and W.D. Sudderth (1999), “Consistency and Strong Inconsistency of Group–Invariant Predictive Inferences”, Bernoulli, 5, pp. 833–854. Eaton, M.L. and W.D. Sudderth (2002), “Group Invariant Inference and Right Haar Measure”, Journal of Statistical Planning and Inference, 103, pp. 87–99. Eaton, M.L. and W.D. Sudderth (2004), “Properties of Right Haar Predictive Inference, Sankhya, 66, pp. 487–512. Eaton, M.L. and W.D. Sudderth (2010), “Invariance of Posterior Distributions Under Reparametrization”, Sankhya, series A, 72, pp. 101–118. Fraser, D.A.S. (1968), The Structure of Inference, John Wiley & Sons, New York. Heath, D. and W.D. Sudderth (1978), “On Finitely Additive Priors, Coherence and Extended Admissibility, The Annals of Statistics, 6, pp. 333–345. Heath, D. and W.D. Sudderth (1989), “Coherent Inference from Improper and from Finitely Additive Priors”, The Annals of Statistics, 17, pp. 907–919. Helland, I.S. (2004), “Statistical Inference under Symmetry”, International Statistical Review, 72, pp. 409–422. Keyes, T.K and M.S. Levy (1996), “Goodness of Prediction Fit for Multivariate Linear Models”, Journal of the American Statistical Association, 91, pp. 191–197. Lane, D.A and W.D. Sudderth (1983), “Coherent and Continuous Inference”, The Annals of Statistics, 11, pp. 114–120. Lehmann, E.L. (1986), Testing Statistical Hypotheses, Second edition, Springer–Verlag, New York. Lehmann, E.L. and G. Casella (1998), Theory of Point Estimation, Second edition, Springer–Verlag, New York. Nachbin, L. (1965), The Haar Integral, D. Van Nostrand Company, Princeton, New Jersey. Pratt, J.W. (1976), “Comment” on “Strong Inconsistency from Uniform Priors” by M. Stone, Journal of the American Statistical Association, 71, pp. 119–120. Stone, M. (1965), “Right Haar Measure for Convergence in Probability to Quasi Posterior Distributions”, Annals of Mathematical Statistics, 36, pp. 440–453. Stone, M. (1976), “Strong Inconsistency from Uniform Priors” with discussion, Journal of the American Statistical Association, 71, pp. 114–125. Sudderth, W.D. (1994), “Coherent Inference and Prediction in Statistics”, in: D. Prawitz, B. Skyrms and D. Westerståhl, eds., Logic, Methodology and Philosophy of Science IX, Elsevier Science, Amsterdam. Villegas, C. (1971), “On Haar Priors”, in: V.P. Godambe and D.A. Sprott, eds., Foundations of Statistical Inference, Holt, Rinehart and Winston, Toronto. Villegas, C. (1981), “Inner Statistical Inference II”, The Annals of Statistics, 9, pp. 768–776. Wijsman, R. A. (1986), “Global Cross Sections as a Tool for Factorization of Measures and Distribution of Maximal Invariants”, Sankhya, series A, 48, pp. 1–42. Wijsman, R. A. (1990), Invariant Measures on Groups and Their Use in Statistics, Institute of Mathematical Statistics Lecture Notes–Monograph Series, vol. 14., Hayward, California. Yang, R. and J.O. Berger (1994), “Estimation of a Covariance Matrix Using the Reference Prior”, The Annals of Statistics, 22, pp. 1195–1211. Zellner, A. (1977), “Maximal Data Information Prior Distributions”, in: A. Aykac and C. Brumat, eds., New Developments in the Applications of Bayesian Methods, North–Holland. Zidek, J.V. (1969), “A Representation of Bayes Invariant Procedures in Terms of Haar Measure”, Annals of the Institute of Statistical Mathematics, 21, pp. 291–308. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/28731 