title: Locally Adaptive Bayes Nonparametric Regression via Nested Gaussian Processes.
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'smth-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.
authors.affiliation: Tenure-Track Principal Investigator, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Rockville, MD 20852.
authors.forename: Bin
authors.initials: B
authors.lastname: Zhu
authors.affiliation: Arts & Sciences Distinguished Professor, Department of Statistical Science, Duke University, Durham, NC 27708.
authors.forename: David B
authors.initials: DB
authors.lastname: Dunson
keywords: Bayesian nonparametric regression; Nested Gaussian processes; Nested smoothing spline; Penalized sum-of-square; Reproducing kernel Hilbert space; Stochastic differential equations