Locally Adaptive Bayes Nonparametric Regression via Nested Gaussian Processes

• Activity Number: 652
• Type: Contributed
• Date/Time: Thursday, August 8, 2013 : 8:30 AM to 10:20 AM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Bin Zhu*+ and David B. Dunson
• Companies: Biostatistics Branch, DCEG, NCI and Duke University
• Keywords: Bayesian nonparametric regression ; Nested Gaussian processes ; Nested smoothing spline ; Penalized sum-of-square ; Reproducing kernel Hilbert space ; Stochastic differential equations

Abstract

We propose a nested Gaussian process (nGP) as a locally adaptive prior for Bayesian nonparametric regression. Specified through a set of stochastic differential equations (SDEs), the nGP imposes a Gaussian process prior for the function's mth-order derivative. The nesting comes in through including a local instantaneous mean function, which is drawn from another Gaussian process inducing adaptivity to locally-varying smoothness. We discuss the support of the nGP prior in terms of the closure of a reproducing kernel Hilbert space, and consider theoretical properties of the posterior. The posterior mean under the nGP prior is shown to be equivalent to the minimizer of a nested penalized sum-of-squares involving penalties for both the global and local roughness of the function. Using highly-efficient Markov chain Monte Carlo for posterior inference, the proposed method performs well in simulation studies compared to several alternatives, and is scalable to massive data, illustrated through a proteomics application.