Abstract
For an invariant statistical model we consider the induced right Haar prior distribution and the resulting right Haar posterior. Using this posterior distribution, HPD (highest posterior density) regions of constant posterior probability are constructed and shown to exhibit probability matching. Further, HPD regions for equivariant functions of the parameter are also shown to exhibit probability matching. A number of examples are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berk, R.H. (1967). A special group structure and equivariant estimation. Ann. Math. Statist., 38, 1436–1445.
Box, G.E.P and Tiao, G.C. (1973). Bayesian inference in statistical analysis. Addison-Wesley, Reading, MA.
Eaton, M.L. (1983, 2007). Multivariate statistics: a vector space approach. Originally published by Wiley, New York. Reprinted by Institute of Mathematical Statistics as vol. 53 in LMNS, Beachwood, Ohio.
Eaton, M.L. (1989). Group invariance applications in statistics. In Regional Conference Series in Probability and Statistics, 1. Institute of Mathematical Statistics, Beachwood, Ohio.
Eaton, M.L. (2008). Dutch book in simple multivariate normal prediction: another look. In Probability and Statistics: Essays in Honor of David A. Freedman. IMS Collections, 2 (D. Nolan and T. Speed, eds.). Institute of Matmematical Statistics, Beachwood, Ohio, pp 12–23.
Eaton, M.L. and Freedman, D.A. (2004). Dutch book against some “objective” priors. Bernoulli, 10, 861–872.
Eaton, M.L., Muirhead, R.J. and Pickering, E.H. (2006). Assessing a vector of clinical observations. J. Statist. Plann. Inference, 136, 3383–3414.
Eaton, M.L. and Sudderth, W.D. (1999). Consistency and strong inconsistency of group invariant predictive inferences. Bernoulli, 5, 833–854.
Eaton, M.L. and Sudderth, W.D. (2001). Best invariant predictive inferences. In Algebraic Methods in Statistics and Probability (M. Viana and D. Richards, eds.). Contemp. Math., 287. American Mathematical Society, Providence, Rhode Island, pp 49–62.
Eaton, M.L. and Sudderth, W.D. (2002). Group invariant inference and right Haar measure. J. Statist. Plann. Inference, 103, 87–99.
Eaton, M.L. and Sudderth, W.D. (2004). Properties of right Haar predictive inference. Sankhyā, 66, 487–512.
Eaton, M.L. and Sudderth, W.D. (2010). Invariance of posterior distributions under reparameterization. Sankhyā, 72-A, 101–118.
Fisher, R.A. (1930). Inverse probability. Proc. Cambridge Philos. Soc., 26, 528–535.
Hooper, P. (1982). Invariant confidence sets with smallest expected measure. Ann. Statist., 10, 1283–1294.
Hora, R.B. and Buehler, R.J. (1966). Fiducial theory and invariant estimation. Ann. Math. Statist., 37, 643–656.
Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proc. R. Soc. Lond. Ser. A, 186, 453–461.
Kiefer, J. (1957). Invariance, minimax sequential estimation, and continuous time processes. Ann. Math. Statist., 28, 573–601.
Lehmann, E.L. (1986). Testing statistical hypotheses, second edition. Springer-Verlag, New York.
Lehmann, E.L. and Casella, G. (1998). Theory of point estimation, second edition. Springer-Verlag, New York.
Nachbin, L. (1965). The Haar integral. Van Nostrand, Princeton, N.J.
Peisakoff, M. (1951). Transformation of Parameters. Unpublished Ph.D. Thesis. Princeton University.
Pitman, E.J.G. (1939). The estimation of the location and scale parameters of a continuous population of any given form. Biometrika, 30, 391–421.
Robert, C.P. (1994). The Bayesian choice. Springer-Verlag, New York.
Severini, T.A., Mukerjee, R. and Ghosh, M. (2002). On an exact probability matching property of right-invariant priors. Biometrika, 89, 952–957.
Stein, C. (1959). An examination of wide discrepancy between fiducial and confidence intervals. Ann. Math. Statist., 30, 877–880.
Stein, C. (1965). Approximation of improper prior measures by prior probability measures. In Bernoulli (1713)-Bayes (1763)-LaPlace (1813), (L.M. LeCam and J. Neyman, eds.). Springer-Verlag, New York, pp 217–240.
Zhu, H. (2002). Invariant Predictive Inferences with Applications. Unpublished Ph.D. Thesis. University of Minnesota.
Acknowledgements
Portions of M. L. Eaton’s research were supported by a grant from the University of Minnesota Retirees Association, Office of the Vice President for Research at the University of Minnesota.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Eaton, M.L., Sudderth, W.D. Invariance, model matching and probability matching. Sankhya A 74, 170–193 (2012). https://doi.org/10.1007/s13171-012-0018-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13171-012-0018-4