The Bayes Lepski's Method and Credible Bands through Volume of Tubular Neighborhoods

Abstract

For a general class of priors based on random series basis expansion, we develop the Bayes Lepski's method to estimate unknown regression function. In this approach, the series truncation point is determined based on a stopping rule that balances the posterior mean bias and the posterior standard deviation. Equipped with this mechanism, we present a method to construct adaptive Bayesian credible bands, where this statistical task is reformulated into a problem in geometry, and the band's radius is computed based on finding the volume of certain tubular neighborhood embedded on a unit sphere. We consider two special cases involving B-splines and wavelets, and discuss some interesting consequences such as the uncertainty principle and self-similarity. Lastly, we show how to program the Bayes Lepski stopping rule on a computer, and numerical simulations in conjunction with our theoretical investigations concur that this is a promising Bayesian uncertainty quantification procedure.

Bayesian mode and maximum estimation and accelerated rates of contraction

“…Coverage of uniform norm credible sets in the adaptive setting may even need a more radical technique (cf. Yoo and van der Vaart [38]).…”

Discussion on Bayesian Cluster Analysis: Point Estimation and Credible Balls by Sara Wade and Zoubin Ghahramani

“…Instead of using infinite mixtures which entails searching over the entire partition space, one can use overfitted mixtures as investigated in Rousseau and Mengersen (2011), where one intentionally overfit the model by choosing a larger but finite number of components than necessary and use some sparsityinducing priors to zero out the unnecessary components. Alternatively, by observing in Table 2 that the number of clusters for the VI credible ball stays constant for the different sample sizes considered, its robust property suggests that we could first try to estimate the correct number of clusters, through MAP (Maximum a posteriori) or the recently proposed Bayes Lepski's method (Yoo and van der Vaart, 2018), and only explore the part of the partition space corresponding to this estimated number of clusters.…”