Evaluating default priors with a generalization of Eaton's Markov chain
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, p. 1069-1091

We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let 𝛷 be a class of functions on the parameter space and consider estimating elements of 𝛷 under quadratic loss. If the formal Bayes estimator of every function in 𝛷 is admissible, then the prior is strongly admissible with respect to 𝛷. Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with the inferential setting. In previous work, this was handled differently depending upon whether ϕ𝛷 was bounded or unbounded. We consider a new Markov chain which allows us to unify and generalize existing approaches while simultaneously broadening the scope of their potential applicability. We use our general theory to investigate strong admissibility conditions for location models when the prior is Lebesgue measure and for the p-dimensional multivariate Normal distribution with unknown mean vector θ and a prior of the form ν(θ 2 )dθ.

Nous considérons l’évaluation de lois a priori impropres dans un cadre Bayésien formel en fonction des conséquences de leur utilisation. Soit 𝛷 une classe de fonctions sur l’espace des paramètres, que l’on cherche à estimer sous une fonction de perte quadratique. Si l’estimateur Bayésien de toute fonction dans 𝛷 est admissible, alors la loi a priori est fortement admissible par rapport à 𝛷. La méthode d’Eaton pour établir l’admissibilité forte est basée sur l’étude des propriétés de stabilité d’une certaine chaîne de Markov associé au cadre inférentiel. Dans des travaux précédents, nous considérions une nouvelle chaîne de Markov qui permet d’unifier et de généraliser les approches existantes tout en élargissant simultanément son champ d’application. Nous utilisons cette théorie générale pour étudier des conditions d’admissibilité forte pour des modéles à paramètre de position, une loi a priori donnée par la mesure de Lebesgue et la loi normale multivariée de dimension p et moyenne θ, et une loi a priori de la forme ν(θ 2 )dθ.

DOI : https://doi.org/10.1214/13-AIHP552
Classification:  62C15,  60J05
Keywords: admissibility, improper prior distribution, symmetric Markov chain, recurrence, Dirichlet form, formal Bayes rule
     author = {Shea, Brian P. and Jones, Galin L.},
     title = {Evaluating default priors with a generalization of Eaton's Markov chain},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {3},
     year = {2014},
     pages = {1069-1091},
     doi = {10.1214/13-AIHP552},
     zbl = {1298.62016},
     mrnumber = {3224299},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_3_1069_0}
Shea, Brian P.; Jones, Galin L. Evaluating default priors with a generalization of Eaton's Markov chain. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 3, pp. 1069-1091. doi : 10.1214/13-AIHP552. http://www.numdam.org/item/AIHPB_2014__50_3_1069_0/

[1] J. Berger and W. E. Strawderman. Choice of hierarchical priors: Admissibility of Normal means. Ann. Statist. 24 (1996) 931-951. | MR 1401831 | Zbl 0865.62004

[2] J. Berger, W. E. Strawderman and D. Tan. Posterior propriety and admissibility of hyperpriors in Normal hierarchical models. Ann. Statist. 33 (2005) 606-646. | MR 2163154 | Zbl 1068.62005

[3] A. C. Brandwein and W. E. Strawderman. Stein estimation for spherically symmetric distributions: Recent developments. Stat. Sci. 27 (2012) 11-23. | MR 2953492 | Zbl 0955.62611

[4] L. D. Brown. Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 (1971) 855-903. | MR 286209 | Zbl 0246.62016

[5] K. L. Chung and W. H. Fuchs. On the distribution of values of sums of random variables. Mem. Amer. Math. Soc. 6 (1951) 1-12. | MR 40610 | Zbl 0042.37502

[6] M. L. Eaton. A method for evaluating improper prior distributions. In Statistical Decision Theory and Related Topics III. S. S. Gupta and J. O. Berger (Eds). Academic Press, Inc., New York, 1982. | MR 705296 | Zbl 0581.62005

[7] M. L. Eaton. A statistical diptych: Admissible inferences - Recurrence of symmetric Markov chains. Ann. Statist. 20 (1992) 1147-1179. | MR 1186245 | Zbl 0767.62002

[8] M. L. Eaton. Admissibility in quadratically regular problems and recurrence of symmetric Markov chains: Why the connection? J. Statist. Plan. Inference 64 (1997) 231-247. | MR 1621615 | Zbl 0944.62010

[9] M. L. Eaton. Markov chain conditions for admissibility in estimation problems with quadratic loss. In State of the Art in Probability and Statistics: Festschrift for Willem R. van Zwet 223-243. M. de Gunst, C. Klaasen and A. van der Vaart (Eds). IMS Lecture Notes Ser. 36. IMS, Beechwood, OH, 2001. | MR 1836563

[10] M. L. Eaton. Evaluating improper priors and recurrence of symmetric Markov chains: An overview. In A Festschrift for Herman Rubin 5-20. A. DasGupta (Ed.). IMS Lecture Notes Ser. 45. IMS, Beechwood, OH, 2004. | MR 2126883 | Zbl 1268.62010

[11] M. L. Eaton, J. P. Hobert and G. L. Jones. On perturbations of strongly admissible prior distributions. Ann. Inst. Henri Poincaré Probab. Stat. 43 (2007) 633-653. | Numdam | MR 2347100 | Zbl 1118.62009

[12] M. L. Eaton, J. P. Hobert, G. L. Jones and W.-L. Lai. Evaluation of formal posterior distributions via Markov chain arguments. Ann. Statist. 36 (2008) 2423-2452. | MR 2458193 | Zbl 1274.62078

[13] J. P. Hobert and C. P. Robert. Eaton’s Markov chain, its conjugate partner, and 𝒫-admissibility. Ann. Statist. 27 (1999) 361-373. | MR 1701115 | Zbl 0945.62012

[14] J. P. Hobert and J. Schweinsberg. Conditions for recurrence and transience of a Markov chain on + and estimation of a geometric success probability. Ann. Statist. 30 (2002) 1214-1223. | MR 1926175 | Zbl 1103.60315

[15] J. P. Hobert, A. Tan and R. Liu. When is Eaton's Markov chain irreducible? Bernoulli 13 (2007) 641-652. | MR 2348744 | Zbl 1131.60066

[16] W. James and C. Stein (1961). Estimation with quadratic loss. In Proc. Fourth Berkeley Symp. Math. Statist. Probab., Vol. 1 361-380. Univ. California Press, Berkeley. | MR 133191 | Zbl 1281.62026

[17] B. W. Johnson. On the admissibility of improper Bayes inferences in fair bayes decision problems. Ph.D. thesis, Univ. Minnesota, 1991. | MR 2686274

[18] R. E. Kass and L. Wasserman. The selection of prior distributions by formal rules. J. Amer. Statist. Assoc. 91 (1996) 1343-1370. | MR 1478684 | Zbl 0884.62007

[19] W.-L. Lai. Admissibility and recurrence of Markov chains with applications. Ph.D. thesis, Univ. Minnesota, 1996.

[20] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer, London, 1993. | MR 1287609 | Zbl 0925.60001

[21] D. Revuz. Markov Chains, 2nd edition. North-Holland, Amsterdam, 1984. | MR 758799 | Zbl 0332.60045

[22] C. Stein. The admissibility of Pitman's estimator of a single location parameter. Ann. Math. Statist. 30 (1959) 970-979. | MR 109392 | Zbl 0087.15101

[23] G. Taraldsen and B. H. Lindqvist. Improper priors are not improper. Amer. Statist. 64 (2010) 154-158. | MR 2757006