Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression
By William Weimin Yoo, Subhashis Ghosal. 2014
89 citations recorded by the source corpus.
In the setting of nonparametric multivariate regression with unknown error variance, we study asymptotic properties of a Bayesian method for estimating a regression function f and its mixed partial derivatives. We use a random series of tensor product of B-splines with normal basis coefficients as a prior for f, and the error variance is either estimated using the empirical Bayes approach or is endowed with a suitable prior in a hierarchical Bayes approach. We establish pointwise, L2 and supremum norm posterior contraction rates for f and its mixed partial derivatives, and show that they coincide with the minimax rates. Our results cover even the anisotropic situation, where the true regression function may have different smoothness in different directions. Using the convergence bounds, we show that pointwise, L2 and supremum norm credible sets for f and its mixed partial derivatives have guaranteed frequentist coverage with optimal size. New results on tensor products of B-splines are also obtained in the course.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c14d9cbe276b45568da51fd97414d180767027f0
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Citations in the corpus
- Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “A relevant work in this regard is [53], where the authors derive the posterior contraction rates under the supremum norm for GP regression induced by tensor products of B-splines, and identify conditions under which the coverage of the simultaneous credible band tends to one as the sample size…”