The Annals of Statistics

Anisotropic function estimation using multi-bandwidth Gaussian processes

Anirban Bhattacharya, Debdeep Pati, and David Dunson

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In nonparametric regression problems involving multiple predictors, there is typically interest in estimating an anisotropic multivariate regression surface in the important predictors while discarding the unimportant ones. Our focus is on defining a Bayesian procedure that leads to the minimax optimal rate of posterior contraction (up to a log factor) adapting to the unknown dimension and anisotropic smoothness of the true surface. We propose such an approach based on a Gaussian process prior with dimension-specific scalings, which are assigned carefully-chosen hyperpriors. We additionally show that using a homogenous Gaussian process with a single bandwidth leads to a sub-optimal rate in anisotropic cases. Zanten (2009) showed that rescaling a homogeneous smooth

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Ann. Statist., Volume 42, Number 1 (2014), 352-381.

First available in Project Euclid: 19 March 2014

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Zentralblatt MATH identifier

Primary: 62G07: Density estimation 62G20: Asymptotic properties
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Adaptive anisotropic Bayesian nonparametrics function estimation Gaussian process rate of convergence


Bhattacharya, Anirban; Pati, Debdeep; Dunson, David. Anisotropic function estimation using multi-bandwidth Gaussian processes. Ann. Statist. 42 (2014), no. 1, 352--381. doi:10.1214/13-AOS1192.

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