Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Posterior Concentration Rates for Bayesian Penalized Splines (2109.04288v3)

Published 9 Sep 2021 in math.ST, stat.ME, and stat.TH

Abstract: Despite their widespread use in practice, the asymptotic properties of Bayesian penalized splines have not been investigated so far. We close this gap and study posterior concentration rates for Bayesian penalized splines in a Gaussian nonparametric regression model. A key feature of the approach is the hyperprior on the smoothing variance, which allows for adaptive smoothing in practice but complicates the theoretical analysis considerably as it destroys conjugacy and precludes analytic expressions for the posterior moments. To derive our theoretical results, we rely on several new concepts including a carefully defined proper version of the partially improper penalized splines prior as well as an innovative spline estimator that projects the observations onto the first basis functions of a Demmler-Reinsch basis. Our results show that posterior concentration at near optimal rate can be achieved if the hyperprior on the smoothing variance strikes a fine balance between oversmoothing and undersmoothing, which can for instance be met by a Weibull hyperprior with shape parameter 1/2. We complement our theoretical results with empirical evidence demonstrating the adaptivity of the hyperprior in practice.

Summary

We haven't generated a summary for this paper yet.