A Direct Approach to Understanding Posterior Consistency of Bayesian Regression Problems

References

• Barron , A. , Schervish , M. J. , Wasserman , L. ( 1999 ). The consistency of posterior distributions in nonparametric problems . Ann. Statist. 27 ( 2 ): 536 – 561 .
   Web of Science ®Google Scholar
• Boos , D. D. ( 1985 ). A converse to Scheffé's theorem . Ann. Statist. 13 ( 1 ): 423 – 427 .
   Google Scholar
• Buck , R. C. ( 1978 ). Advanced Calculus. , 3rd ed. New York : McGraw-Hill Book Co. With the collaboration of Ellen F. Buck, International Series in Pure and Applied Mathematics .
   Google Scholar
• Choi , T. , Lee , J. , Roy , A. ( 2009 ). A note on the Bayes factor in a semiparametric regression model . J. Multivariate Anal. 100 ( 6 ): 1316 – 1327 .
   Google Scholar
• Choi , T. , Ramamoorthi , R. V. ( 2008 ). Remarks on consistency of posterior distributions. In: Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, Vol. 3 of Institute of Mathematical Statistics Collection, Institute of Mathematical Statistics, Beachwood, OH: pp. 170–186 .
   Google Scholar
• Choi , T. , Schervish , M. J. ( 2007 ). On posterior consistency in nonparametric regression problems . J. Multivariate Anal. 98 ( 10 ): 1969 – 1987 .
   Web of Science ®Google Scholar
• Eubank , R. L. , Spiegelman , C. H. ( 1990 ). Testing the goodness of fit of a linear model via nonparametric regression techniques . J. Amer. Statist. Assoc. 85 ( 410 ): 387 – 392 .
   Web of Science ®Google Scholar
• Ghosal , S. ( 1999 ). Asymptotic normality of posterior distributions in high-dimensional linear models . Bernoulli 5 ( 2 ): 315 – 331 .
   Google Scholar
• Ghosal , S. , Roy , A. ( 2006 ). Posterior consistency of Gaussian process prior for nonparametric binary regression . Ann. Statist. 34 ( 5 ): 2413 – 2429 .
   Web of Science ®Google Scholar
• Ghosal , S. , van der Vaart , A. (2007). Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Statist. 35(1):192–223.
   Web of Science ®Google Scholar
• Ghosal , S. , Ghosh , J. K. , Ramamoorthi , R. V. ( 1999 ). Posterior consistency of Dirichlet mixtures in density estimation . Ann. Statist. 27 ( 1 ): 143 – 158 .
   Web of Science ®Google Scholar
• Ghosh , J. K. , Ramamoorthi , R. V. ( 2003 ). Bayesian Nonparametrics . Springer Series in Statistics. New York : Springer-Verlag .
   Google Scholar
• Lenk , P. J. ( 1999 ). Bayesian inference for semiparametric regression using a Fourier representation . J. Roy. Statist. Soc. Ser. B Statist Methodol. 61 ( 4 ): 863 – 879 .
   Web of Science ®Google Scholar
• Lindley , D. V. , Smith , A. F. M. ( 1972 ). Bayes estimates for the linear model . J. Roy. Statist. Soc. Ser. B 34 : 1 – 41 .
   Google Scholar
• Rencher , A. C. , Schaalje , G. B. ( 2008 ). Linear Models in Statistics. , 2nd ed. Hoboken , NJ : Wiley-Interscience .
   Google Scholar
• Scheffé , H. ( 1947 ). A useful convergence theorem for probability distributions . Ann. Math. Statist. 18 : 434 – 438 .
   Google Scholar
• Scricciolo , C. ( 2008 ). A note on Bayesian nonparametric regression function estimation . Statist. Meth. Appl. 17 ( 3 ): 321 – 334 .
   Google Scholar
• Shen , X. , Wasserman , L. ( 2001 ). Rates of convergence of posterior distributions . Ann. Statist. 29 ( 3 ): 687 – 714 .
   Web of Science ®Google Scholar
• Walker , S. ( 2003 ). Bayesian consistency for a class of regression problems . South African Statist. J. 37 ( 2 ): 151 – 169 .
   Google Scholar
• Walker , S. ( 2004 ). New approaches to Bayesian consistency . Ann. Statist. 32 ( 5 ): 2028 – 2043 .
   Web of Science ®Google Scholar
• Zhao , L. H. ( 2000 ). Bayesian aspects of some nonparametric problems . Ann. Statist. 28 ( 2 ): 532 – 552 .
   Google Scholar