An Analysis of Bayesian Inference for Nonparametric Regression

Cox, Dennis D.

Ann. Statist., Tome 21 (1993) no. 1, p. 903-923 / Harvested from Project Euclid

Résumé

The observation model $y_i = \beta(i/n) + \varepsilon_i, 1 \leq i \leq n$, is considered, where the $\varepsilon$'s are i.i.d. with mean zero and variance $\sigma^2$ and $\beta$ is an unknown smooth function. A Gaussian prior distribution is specified by assuming $\beta$ is the solution of a high order stochastic differential equation. The estimation error $\delta = \beta - \hat{\beta}$ is analyzed, where $\hat{\beta}$ is the posterior expectation of $\beta$. Asymptotic posterior and sampling distributional approximations are given for $\|\delta\|^2$ when $\|\cdot\|$ is one of a family of norms natural to the problem. It is shown that the frequentist coverage probability of a variety of $(1 - \alpha)$ posterior probability regions tends to be larger than $1 - \alpha$, but will be infinitely often less than any $\varepsilon > 0$ as $n \rightarrow \infty$ with prior probability 1. A related continuous time signal estimation problem is also studied.

Détail

Publié le : 1993-06-14
Classification:  Bayesian inference,  nonparametric regression,  confidence regions,  signal extraction,  smoothing splines,  62A15,  62G15,  62J99,  62E20,  62M99,  60G35