Consistency of bayes estimators of a binary regression function

Annals of Statistics

Volumn 34, Issue 3, 2006, Pages 1233-1269

  Coram, Marc ^a   Lalley, Steven P ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

Bayes procedure; Binary regression; Consistency; Large deviations; Sub additivity

Indexed keywords

EID: 33747185043     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053606000000236     Document Type: Article

References (27)

1. BARRON, A., SCHERVISH, M. J. and WASSERMAN, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536-561. MR1714718
2. CHI, Z. (2001). Stochastic sub-additivity approach to the conditional large deviation principle. Ann. Probab. 29 1303-1328. MR1872744
3. CORAM, M. (2002). Nonparametric Bayesian classification. Ph.D. dissertation, Stanford Univ.
4. DIACONIS, P. and FREEDMAN, D. (1986). On inconsistent Bayes estimates of location. Ann. Statist. 14 68-87. MR0829556
5. DIACONIS, P. and FREEDMAN, D. A. (1993). Nonparametric binary regression: A Bayesian approach. Ann. Statist. 21 2108-2137. MR 1245784
6. DIACONIS, P. and FREEDMAN, D. A. (1995). Nonparametric binary regression with random covariates. Probab. Math. Statist. 15 243-273. MR 1369802
7. FREEDMAN, D. A. (1963). On the asymptotic behavior of Bayes' estimates in the discrete case. Ann. Math. Statist. 34 1386-1403. MR0158483
8. FREEDMAN, D. and DIACONIS, P. (1983). On inconsistent Bayes estimates in the discrete case. Ann. Statist. 11 1109-1118. MR0720257
9. GHOSAL, S., GHOSH, J. K. and VAN DER VAART, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. MR1790007
10. GHOSH, J. K. and RAMAMOORTHI, R. V. (2003). Bayesian Nonparametrics. Springer, New York. MR 1992245
11. HAMMERSLEY, J. M. (1962). Generalization of the fundamental theorem on sub-additive functions. Proc. Cambridge Philos. Soc. 58 235-238. MR0137800
12. KAIJSER, T. (1975). A limit theorem for partially observed Markov chains. Ann. Probab. 3 677-696. MR0383536
13. KARLSSON, A. and MARGULIS, G. A. (1999). A multiplicative ergodic theorem and nonpositively curved spaces. Comm. Math. Phys. 208 107-123. MR1729880
14. KIEFFER, J. C. (1973). A counterexample to Perez's generalization of the Shannon-McMillan theorem. Ann. Probab. 1 362-364. MR0351626
15. KIEFFER, J. C. (1974). A simple proof of the Moy-Perez generalization of the Shannon-McMillan theorem. Pacific J. Math. 51 203-206. MR0347448
16. KINGMAN, J. F. C. (1973). Subadditive ergodic theory (with discussion). Ann. Probab. 1 883-909. MR0356192
17. LIGGETT, T. M. (1985). An improved subadditive ergodic theorem. Ann. Probab. 13 1279-1285. MR0806224
18. PEREZ, A. (1964). Extensions of Shannon-McMillan's limit theorem to more general stochastic processes. In Trans. Third Prague Conference on Information Theory, Statistical Decision Functions, Random Processes (Liblice, 1962) 545-574. Publ. House Czech. Acad. Sci., Prague. MR0165996
19. PEREZ, A. (1980). On Shannon-McMillan's limit theorem for pairs of stationary random processes. Kybernetika (Prague) 16 301-314. MR0591959
20. POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York. MR0762984
21. RUELLE, D. (1999). Statistical Mechanics. Rigorous Results. World Scientific, River Edge, NJ. MR1747792
22. SCHWARTZ, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10-26. MR0184378
23. SHEN, X. and WASSERMAN, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687-714. MR1865337
24. STEIN, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press. MR0290095
25. WALKER, S. G. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028-2043. MR2102501
26. WALKER, S. G. (2004). Modern Bayesian asymptotics. Statist. Sci. 19 111-117. MR2082150
27. WALTERS, P. (1982). An Introduction to Ergodic Theory. Springer, New York. MR0648108