Reference priors for exponential families with increasing dimension

Authors

Bertrand Clarke, University of MiamiFollow
Subhashis Ghosal, North Carolina State UniversityFollow

Document Type

Article

Date of this Version

12-2010

Citation

2010 Clarke, Ghosal

Comments

Electronic Journal of Statistics Vol. 4 (2010) 737–780 ISSN: 1935-7524 DOI: 10.1214/10-EJS569

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 = pm 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 Jeffrey's 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.