Dutch book in simple multivariate normal prediction: Another look
Morris L. Eaton
Abstract
In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions.
First Page:
Show
Hide
Primary Subjects: 62H99
Secondary Subjects: 62A01
Keywords: Dutch book; multivariate normal; prediction
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.imsc/1207580075
Digital Object Identifier: doi:10.1214/193940307000000356
References
[1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR233396
[2] Bjørnstad, J. (1990). Predictive likelihood: A review. Statist. Sci. 5 242–254.
[3] de Finetti, B. (1931). Sul significato soggestivo della probabilità. Fundamenta Mathematicae 17 289–329.
[4] de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7 1–68. English translation in Studies in Subjective Probability (1980). H. E. Kyburg, H. E. Smokler, eds. Krieger Publishing, Huntington, NY.
Mathematical Reviews (MathSciNet):
MR584256
[5] Dunford, N. and Schwartz, J. (1957). Linear Operators, Part I. Interscience, New York.
[6] Eaton, M. (1983). Multivariate Statistics. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR716321
[7] Eaton, M. (1989). Invariance applications in statistics. In Regional Conference Series in Probability and Statistics I. Institute of Mathematical Statistics, Beachwood, OH.
[8] Eaton, M. and Freedman, D. (2004). Dutch book against some objective priors. Bernoulli 10 861–872.
Mathematical Reviews (MathSciNet):
MR2093614
Digital Object Identifier: doi:10.3150/bj/1099579159
Project Euclid: euclid.bj/1099579159
[9] Eaton, M. and Sudderth, W. (1993). Prediction in a multivariate normal setting: Coherence and incoherence. Sankhyā Ser. A 55 481–493.
Mathematical Reviews (MathSciNet):
MR1323401
[10] Eaton, M. and Sudderth, W. (1995). The formal posterior of a standard flat prior in MANOVA is coherent. J. Italian Statist. Soc. 2 251–270.
[11] Eaton, M. and Sudderth, W. (1998). A new predictive distribution for normal multivariate linear models. Sankhyā Ser. A 60 363–382.
Mathematical Reviews (MathSciNet):
MR1718856
[12] Eaton, M. and Sudderth, W. (1999). Consistency and strong inconsistency of group invariant predictive inferences. Bernoulli 5 833–854.
Mathematical Reviews (MathSciNet):
MR1715441
Digital Object Identifier: doi:10.2307/3318446
Project Euclid: euclid.bj/1171290401
[13] Eaton, M. and Sudderth, W. (2004). Properties of right Haar predictive inference. Sankhyā Ser. A 66 487–512.
Mathematical Reviews (MathSciNet):
MR2108204
[14] Freedman, D. and Purves, R. (1969). Bayes method for bookies. Ann. Math. Statist. 40 1177–1186.
Mathematical Reviews (MathSciNet):
MR240914
Zentralblatt MATH:
0212.50602
Digital Object Identifier: doi:10.1214/aoms/1177697494
Project Euclid: euclid.aoms/1177697494
[15] Heath, D. and Sudderth, W. (1978). On finitely additive priors, coherence and extended admissibility. Ann. Statist. 6 333–345.
Mathematical Reviews (MathSciNet):
MR464450
Zentralblatt MATH:
0385.62005
Digital Object Identifier: doi:10.1214/aos/1176344128
Project Euclid: euclid.aos/1176344128
[16] Keyes, T. and Levy, M. (1996). Goodness of prediction fit for multivariate linear models. J. Amer. Statist. Assoc. 91 191–197.
Mathematical Reviews (MathSciNet):
MR1394073
Zentralblatt MATH:
0870.62025
Digital Object Identifier: doi:10.2307/2291395
JSTOR: links.jstor.org
[17] Laplace, P. (1774). Mémoire sur la probabilité des causes par les évènemens. Mémoires de mathématique et de physique presentés à l’Academie royal des sciences, par divers savans, & lûs dans ses assemblées 6 621–656. (Reprinted in Laplaces’s Oeuvres Complétes 8 27–65).
[18] Ramsay, F. (1931). Truth and probability. In F. P. Ramsay, The Foundations of Mathematics and other Logical Essays Chapter VII (R. Braithwaite, ed.) 156–198. Trubner & Co., London.
[19] Stigler, S. (1986). Laplace’s 1774 Memoir on Inverse Probability. Statist. Sci. 1 359–378.
Mathematical Reviews (MathSciNet):
MR858515
Digital Object Identifier: doi:10.1214/ss/1177013620
Project Euclid: euclid.ss/1177013620
[20] Stone, M. (1976). Strong inconsistency from uniform priors. J. Amer. Statist. Assoc. 71 114–125.
Mathematical Reviews (MathSciNet):
MR415866
Zentralblatt MATH:
0335.62026
Digital Object Identifier: doi:10.2307/2285744
JSTOR: links.jstor.org
[21] Zhu, H. (2002). Invariant predctive inferences with applications. Ph.D. thesis, Univ. Minnesota.
Mathematical Reviews (MathSciNet):
MR2703975
Institute of Mathematical Statistics Collections
