Asymptotic Normality of Posterior Distributions for Exponential Families when the Number of Parameters Tends to Infinity

• Published in: Journal of multivariate analysis Vol. 74; no. 1; pp. 49 - 68
• Main Author: Ghosal, Subhashis
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.07.2000; Elsevier
• Series: Journal of Multivariate Analysis
• Subjects: Exact sciences and technology; exponential family; exponential family, normal approximation, posterior consistency, posterior distribution; Linear inference, regression; Mathematics; normal approximation; Parametric inference; posterior consistency; posterior distribution; Probability and statistics; Sciences and techniques of general use; Statistics; exponential family, normal approximation, posterior consistency, posterior distribution; Bayes estimation; Parameter estimation; Growth rate; Prior distribution; Riemann metric; Consistency; Gaussian distribution; Chi square; Fisher information; Lipschitz continuity; Euclidian norm; Asymptotic normality; Positive definite matrix; Tchebychev inequality; Exponential family; Density function; Multinomial distribution; Maximum likelihood; Posterior distribution
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1006/jmva.1999.1874

We study consistency and asymptotic normality of posterior distributions of the natural parameter for an exponential family when the dimension of the parameter grows with the sample size. Under certain growth restrictions on the dimension, we show that the posterior distributions concentrate in neighbourhoods of the true parameter and can be approximated by an appropriate normal distribution.