Objective priors for the bivariate normal model. (English) Zbl 1133.62014

Berger, James O.; Sun, Dongchu

Ann. Stat. 36, No. 2, 963-982 (2008).

Summary: Study of the bivariate normal distribution raises the full range of issues involving objective Bayesian inference, including the different types of objective priors (e.g., Jeffreys, invariant, reference, matching), the different modes of inference (e.g., Bayesian, frequentist, fiducial) and the criteria involved in deciding on optimal objective priors (e.g., ease of computation, frequentist performance, marginalization paradoxes). Summary recommendations as to optimal objective priors are made for a variety of inferences involving the bivariate normal distribution. In the course of the investigation, a variety of surprising results were found, including the availability of objective priors that yield exact frequentist inferences for many functions of the bivariate normal parameters, including the correlation coefficient.

MSC:

62F15	Bayesian inference
62F10	Point estimation
62A01	Foundations and philosophical topics in statistics
62F25	Parametric tolerance and confidence regions
62H12	Estimation in multivariate analysis
62H10	Multivariate distribution of statistics

Keywords:

reference priors; matching priors; Jeffreys priors; right-Haar prior; fiducial inference; frequentist coverage; marginalization paradox; rejection sampling; constructive posterior distributions

References:

[1]	Bayarri, M. J. (1981). Inferencia bayesiana sobre el coeficiente de correlación de una población normal bivariante. Trabajos de Estadistica e Investigacion Operativa 32 18-31.
[2]	Bayarri, M. J. and Berger, J. (2004). The interplay between Bayesian and frequentist analysis. Statist. Sci. 19 58-80. · Zbl 1062.62001 · doi:10.1214/088342304000000116
[3]	Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors (with discussion). In Bayesian Statistics 4 35-60. Oxford Univ. Press. · Zbl 1045.62502
[4]	Berger, J. O., Strawderman, W. and Tang, D. (2005). Posterior propriety and admissibility of hyperpriors in normal hierarchical models. Ann. Statist. 33 606-646. · Zbl 1068.62005 · doi:10.1214/009053605000000075
[5]	Berger, J. O. and Sun, D. (2006). Objective priors for a bivariate normal model with multivariate generalizations. Technical Report 07-06, ISDS, Duke Univ.
[6]	Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). J. Roy. Statist. Soc. Ser. B 41 113-147. JSTOR: · Zbl 0428.62004
[7]	Brillinger, D. R. (1962). Examples bearing on the definition of fiducial probability with a bibliography. Ann. Math. Statist. 33 1349-1355. · Zbl 0217.50901 · doi:10.1214/aoms/1177704367
[8]	Brown, P., Le, N. and Zidek, J. (1994). Inference for a covariance matrix. In Aspects of Uncertainty : A Tribute to D. V. Lindley (P. R. Freeman and A. F. M. Smith, eds.) 77-92. Wiley, Chichester. · Zbl 0839.62019
[9]	Datta, G. and Mukerjee, R. (2004). Probability Matching Priors : Higher Order Asymptotics . Springer, New York. · Zbl 1044.62031
[10]	Datta, G. S. and Ghosh, J. K. (1995a). On priors providing frequentist validity for Bayesian inference. Biometrika 82 37-45. JSTOR: · Zbl 0823.62004 · doi:10.2307/2337625
[11]	Datta, G. S. and Ghosh, J. K. (1995b). Noninformative priors for maximal invariant parameter in group models. Test 4 95-114. · Zbl 0851.62002 · doi:10.1007/BF02563105
[12]	Datta, G. S. and Ghosh, M. (1995c). Some remarks on noninformative priors. J. Amer. Statist. Assoc. 90 1357-1363. JSTOR: · Zbl 0878.62003 · doi:10.2307/2291526
[13]	Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference (with discussion). J. Roy. Statist. Soc. Ser. B 35 189-233. JSTOR: · Zbl 0271.62009
[14]	Fisher, R. A. (1930). Inverse probability. Proc. Cambridge Philos. Soc. 26 528-535. · JFM 56.1083.05
[15]	Fisher, R. A. (1956). Statistical Methods and Scientific Inference . Oliver and Boyd, Edinburgh. · Zbl 0070.36903
[16]	Geisser, S. and Cornfield, J. (1963). Posterior distributions for multivariate normal parameters. J. Roy. Statist. Soc. Ser. B 25 368-376. JSTOR: · Zbl 0124.35304
[17]	Ghosh, M. and Yang, M.-C. (1996). Noninformative priors for the two sample normal problem. Test 5 145-157. · Zbl 0852.62002 · doi:10.1007/BF02562686
[18]	Lehmann, E. L. (1986). Testing Statistical Hypotheses , 2nd ed. Wiley, New York. · Zbl 0608.62020
[19]	Lindley, D. V. (1961). The use of prior probability distributions in statistical inference and decisions. Proc. 4th Berkeley Sympos. Math. Statist. Probab. 1 (J. Neyman and E. L. Scott, eds.) 453-468. Univ. California Press, Berkeley. · Zbl 0109.36901
[20]	Lindley, D. V. (1965). Introduction to Probability and Statistics from a Bayesian Viewpoint . Cambridge Univ. Press. · Zbl 0123.34505
[21]	Pratt, J. W. (1963). Shorter confidence intervals for the mean of a normal distribution with known variance. Ann. Math. Statist. 34 574-586. · Zbl 0114.35502 · doi:10.1214/aoms/1177704170
[22]	Rao, C. R. (1973). Linear Statistical Inference and Its Applications . Wiley, New York. · Zbl 0256.62002
[23]	Severini, T. A., Mukerjee, R. and Ghosh, M. (2002). On an exact probability matching property of right-invariant priors. Biometrika 89 952-957. JSTOR: · Zbl 1034.62022 · doi:10.1093/biomet/89.4.952
[24]	Stone, M. and Dawid, A. P. (1972). Un-Bayesian implications of improper Bayes inference in routine statistical problems. Biometrika 59 369-375. JSTOR: · Zbl 0239.62004 · doi:10.1093/biomet/59.2.369
[25]	Yang, R. and Berger, J. (2004). Estimation of a covariance matrix using the reference prior. Ann. Statist. 22 1195-1211. · Zbl 0819.62013 · doi:10.1214/aos/1176325625