Electronic Journal of Statistics

Reference priors for exponential families with increasing dimension

Bertrand Clarke and Subhashis Ghosal

Full-text: Open access

Abstract

In this article, we establish the asymptotic normality of the posterior distribution for the natural parameter in an exponential family based on independent and identically distributed data. The mode of convergence is expected Kullback-Leibler distance and the number of parameters p is increasing with the sample size n. Using this, we give an asymptotic expansion of the Shannon mutual information valid when p=pn increases at a sufficiently slow rate. The second term in the asymptotic expansion is the largest term that depends on the prior and can be optimized to give Jeffreys’ prior as the reference prior in the absence of nuisance parameters. In the presence of nuisance parameters, we find an analogous result for each fixed value of the nuisance parameter. In three examples, we determine the rates at which pn can be allowed to increase while still retaining asymptotic normality and the reference prior property.

Article information

Source
Electron. J. Statist., Volume 4 (2010), 737-780.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1282053980

Digital Object Identifier
doi:10.1214/10-EJS569

Mathematical Reviews number (MathSciNet)
MR2678969

Zentralblatt MATH identifier
1329.62120

Subjects
Primary: 62F15: Bayesian inference
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures

Keywords
Objective prior posterior normality mutual information increasing dimension exponential family

Citation

Clarke, Bertrand; Ghosal, Subhashis. Reference priors for exponential families with increasing dimension. Electron. J. Statist. 4 (2010), 737--780. doi:10.1214/10-EJS569. https://projecteuclid.org/euclid.ejs/1282053980


Export citation

References

  • Berger, J. O. and J. M. Bernardo (1989). Estimating a product of means: Bayesian analysis with reference priors., J. Amer. Statist. Assoc. 84, 200–207.
  • Berger, J. O. and J. M. Bernardo (1991). Reference priors in a variance components problem. In P. Goel and N. Iyengar (Eds.), Bayesian Inference in Statistics and Econometrics, pp. 177–194. New York: Springer.
  • Berger, J. O. and J. M. Bernardo (1992a). On the development of reference priors. In J. M. Bernardo, J. O. Berger, A. Dawid, and A. Smith (Eds.), Bayesian Statistics IV, pp. 36–60. Oxford: Clarendon Press.
  • Berger, J. O. and J. M. Bernardo (1992b). Ordered group reference priors with application to the multinomial., Biometrika 25, 25–37.
  • Berger, J. O., J. M. Bernardo, and M. Mendoza (1991). On priors that maximize expected information. In J. Klein and J. Lee (Eds.), Recent Developments in Statistics and Their Applications, pp. 1–20. Seoul: Freedom Academy.
  • Berger, J. O., J. M. Bernardo, and D. Sun (2009). The formal definition of reference priors., Ann. Statist. 37, 905–938.
  • Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference., J. Roy. Statist. Soc. B 41, 113–147.
  • Bernardo, J. M. (2010). Integrated objective Bayesian estimation and hypothesis testing. In J. M. Bernardo, J. O. Berger, A. P. D. Dawid, and A. F. M. Smith (Eds.), Bayesian Statistics IX, Oxford. Clarendon Press.
  • Boucheron, S. and E. Gassiat (2009). A Bernstein-von Mises theorem for discrete probability distributions., Elec. J. Statist. 3, 114–148.
  • Brown, L. D. (1986)., Fundamentals of Statistical Exponential Families. Vol. 9, Lecture Notes –Monograph Series. Hayward, CA: Institute of Mathematical Statistics.
  • Chen, M.-H., J. Ibrahim, and S. Kim (2009). Properties and implementation of Jeffreys’ prior in binomial regression models., J. Amer. Stat. Assoc. 103, 1659–1664.
  • Clarke, B. and A. Barron (1990). Information-theoretic asymptotics of Bayes methods., IEEE Trans. Inform. Theory 36, 453–471.
  • Clarke, B. and A. Barron (1994). Jeffreys’ prior is the reference prior under entropy loss., J. Stat. Planning and Inference 41, 37–60.
  • Clarke, B. and D. Sun (1997). Reference priors under the chi-square distance., Sankhya 59, 215–231.
  • Clarke, B. and A. Yuan (2004). Partial information reference priors: derivation and interpretations., J. Stat. Plann. Inf. 123, 313–345.
  • Geisser, S. and J. Cornfield (1963). Posterior distributions for multivariate normal parameters., J. Roy. Stat. Soc. Ser. B 25, 368–376.
  • Gelman, A., J. Carlin, S. Stern, and D. Rubin (2004)., Bayesian Data Analysis. Boca Raton, FL: Chapman and Hall.
  • George, E. and R. McCulloch (1993). On obtaining invariant prior distributions., J. Statist. Plann. Inf. 37, 169–179.
  • Ghosal, S. (1997). Normal approximation to the posterior distribution for generalized linear models with many covariates., Math. Methods Statist. 6, 332–348.
  • Ghosal, S. (1999). Asymptotic normality of posterior distributions in high dimensional linear models., Bernoulli 5, 315–331.
  • Ghosal, S. (2000). Asymptotic normality of posterior distributions for exponential families when the number of parameters tends to infinity., J. Multivariate Anal. 74, 49–68.
  • Ghosal, S., J. K. Ghosh, and R. V. Ramamoorthi (1997). Non-informative priors via sieves and packing numbers. In S. Panchapakesan and N. Balakrishnan (Eds.), Advances in Statistical Decision Theory and Applications, pp. 119–132. New York: Birkhauser.
  • Ghosal, S., J. K. Ghosh, and A. W. van der Vaart (2000). Convergence rates of posterior distributions., Ann. Statist. 30 (2), 500–531.
  • Ghosh, J. K. and R. Mukerjee (1992). Noninformative priors. In J. M. Bernardo, J. O. Berger, A. P. D. Dawid, and A. F. M. Smith (Eds.), Bayesian Statistics IV, Oxford, pp. 195–210. Clarendon Press.
  • Ghosh, J. K. and R. V. Ramamoorthi (2003)., Bayesian Nonparametrics. New York, NY: Springer.
  • Ghosh, M., V. Mergel, and R. Liu (2010). A general divergence criterion for prior selection., To appear: Ann. Inst. Stat. Math. .
  • Guan, Y. and J. Dy (2009). Sparse probabilistic principal component analysis. In, JMLR Workshop and Conference Proceedings Vol. 5: AISTATS, pp. 185–192.
  • Heo, T. and J. Kim (2007). Bayesian inference for multinomial group testing., Korean Communications in Statistics 14, 81–92.
  • Ibragimov, I. and R. Hasminsky (1973). On the information in a sample about a parameter. In, Proc. 2nd Internat. Symp. on Information Theory, Budapest, pp. 295–309. Akademiai, Kiado.
  • Lindley, D. (1956). On a measure of the information provided by an experiment., Ann. Math. Statist. 27, 986–1005.
  • Ortega, J. and W. Rheinboldt (1970)., Iterative Solution of Nonlinear Equations in Several Variables. New York, NY: Academic Press.
  • Portnoy, S. (1988). Asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity., Ann. Statist. 16, 356–366.
  • Shannon, C. (1948a). A mathematical theory of communication, part i., Bell Syst. Tech. J. 27, 379 – 423.
  • Shannon, C. (1948b). A mathematical theory of communication, part ii., Bell Syst. Tech. J 27, 623 – 656.
  • Sono, S. (1983). On a non-informative prior distribution for Bayesian inference of multinomial distribution parameters., Ann. Inst. Statist. Math. 35 (Part A), 167–174.
  • Sun, D. and J. O. Berger (1998). Reference priors with partial information., Biometrika 85, 55–71.
  • Yang, R. and J. O. Berger (1994). Estimation of a covariance matrix using a reference prior., Ann. Statist. 22, 1195–1211.
  • Zhang, Z. (1994)., Discrete Noninformative Priors. Ph. D. thesis, Department of Statistics, Yale.
  • Zhu, M. and A. Lu (2004). The counter-intuitive non-informative prior for the Bernoulli family., J. Stat. Ed. 12, 1–10.