The Annals of Statistics
- Ann. Statist.
- Volume 31, Number 2 (2003), 536-559.
Adaptive Bayesian inference on the mean of an infinite-dimensional normal distribution
We consider the problem of estimating the mean of an infinite-break dimensional normal distribution from the Bayesian perspective. Under the assumption that the unknown true mean satisfies a "smoothness condition," we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators. Although the posterior distribution can achieve the optimal rate of convergence, the required prior depends on a "smoothness parameter" q. When this parameter q is unknown, besides the estimation of the mean, we encounter the problem of selecting a model. In a Bayesian approach, this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model. We show that if q takes values only in a discrete set, the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model. A slightly weaker result is presented when q is unrestricted. An adaptive point estimator based on the posterior distribution is also constructed.
Ann. Statist., Volume 31, Number 2 (2003), 536-559.
First available in Project Euclid: 22 April 2003
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 62G20: Asymptotic properties
Secondary: 62C10: Bayesian problems; characterization of Bayes procedures 62G05: Estimation
Belitser, Eduard; Ghosal, Subhashis. Adaptive Bayesian inference on the mean of an infinite-dimensional normal distribution. Ann. Statist. 31 (2003), no. 2, 536--559. doi:10.1214/aos/1051027880. https://projecteuclid.org/euclid.aos/1051027880