Electronic Journal of Statistics

Adaptive estimation of multivariate functions using conditionally Gaussian tensor-product spline priors

R. de Jonge and J.H. van Zanten

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We investigate posterior contraction rates for priors on multivariate functions that are constructed using tensor-product B-spline expansions. We prove that using a hierarchical prior with an appropriate prior distribution on the partition size and Gaussian prior weights on the B-spline coefficients, procedures can be obtained that adapt to the degree of smoothness of the unknown function up to the order of the splines that are used. We take a unified approach including important nonparametric statistical settings like density estimation, regression, and classification.

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Electron. J. Statist. Volume 6 (2012), 1984-2001.

First available in Project Euclid: 30 October 2012

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Zentralblatt MATH identifier

Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 62G20: Asymptotic properties

Nonparametric Bayes procedure tensor-product splines posterior contraction rate adaptive estimation


de Jonge, R.; van Zanten, J.H. Adaptive estimation of multivariate functions using conditionally Gaussian tensor-product spline priors. Electron. J. Statist. 6 (2012), 1984--2001. doi:10.1214/12-EJS735. http://projecteuclid.org/euclid.ejs/1351603386.

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  • [1] Carl de Boor., A practical guide to splines, volume 27 of Applied Mathematical Sciences. Springer-Verlag, New York, revised edition, 2001.
  • [2] R. de Jonge and J.H. van Zanten. Adaptive nonparametric Bayesian inference using location-scale mixture priors., Ann. Statist., 38(6) :3300–3320, 2010.
  • [3] D.G.T. Denison, B.K. Mallick, and A.F.M. Smith. Automatic Bayesian curve fitting., J. R. Statist. Soc. B, 60(60):333–350, 1998.
  • [4] I. DiMatteo, C.R. Genovese, and R.E. Kass. Bayesian curve-fitting with free-knot splines., Biometrika, 88(4) :1055–1071, 2001.
  • [5] Subhashis Ghosal. Convergence rates for density estimation with Bernstein polynomials., Ann. Statist., 29(5) :1264–1280, 2001.
  • [6] Subhashis Ghosal, Jayanta K. Ghosh, and Aad W. van der Vaart. Convergence rates of posterior distributions., Ann. Statist., 28(2):500–531, 2000.
  • [7] Subhashis Ghosal, Jüri Lember, and Aad Van Der Vaart. Nonparametric Bayesian model selection and averaging., Electronic Journal of Statistics, 2:63–89, 2008.
  • [8] Subhashis Ghosal and Aad W. van der Vaart. Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities., Ann. Statist., 29(5) :1233–1263, 2001.
  • [9] Subhashis Ghosal and Aad W. van der Vaart. Convergence rates for posterior distributions for noniid observations., Ann. Statist., 35:697–723, 2007.
  • [10] Tzee-Ming Huang. Convergence rates for posterior distributions and adaptive estimation., Ann. Statist., 32(4) :1556–1593, 2004.
  • [11] Willem Kruijer, Judith Rousseau, and Aad van der Vaart. Adaptive Bayesian density estimation with location-scale mixtures., Electron. J. Stat., 4 :1225–1257, 2010.
  • [12] L. Panzar and J.H. van Zanten. Nonparametric Bayesian inference for ergodic diffusions., J. Statist. Plann. Inference, 139 :4204–4210, 2009.
  • [13] Sonia Petrone and Larry Wasserman. Consistency of Bernstein polynomial posteriors., J. R. Stat. Soc. Ser. B Stat. Methodol., 64(1):79–100, 2002.
  • [14] David Pollard., Empirical processes: theory and applications. NSF-CBMS Regional Conference Series in Probability and Statistics, 2. Institute of Mathematical Statistics, Hayward, CA, 1990.
  • [15] V. Rivoirard and J. Rousseau. Posterior concentration rates for infinite dimensional exponential families., Bayesian Anal., to appear, 2012.
  • [16] Judith Rousseau. Rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density., Ann. Statist., 38(1):146–180, 2010.
  • [17] Larry L. Schumaker., Spline functions: Basic theory. John Wiley & Sons Inc., New York, 1981. Pure and Applied Mathematics, A Wiley-Interscience Publication.
  • [18] W. Shen and S. Ghosal. MCMC-free adaptive Bayesian procedures using random series prior. Preprint, 2012.
  • [19] Xiaotong Shen and Larry Wasserman. Rates of convergence of posterior distributions., Ann. Statist., 29(3):687–714, 2001.
  • [20] Charles J. Stone. Large-sample inference for log-spline models., Ann. Statist., 18(2):717–741, 1990.
  • [21] Charles J. Stone. The use of polynomial splines and their tensor products in multivariate function estimation., Ann. Statist., 22(1):118–184, 1994. With discussion by Andreas Buja and Trevor Hastie and a rejoinder by the author.
  • [22] F.H. van der Meulen, A.W. van der Vaart, and J.H. van Zanten. Convergence rates of posterior distributions for Brownian semimartingale models., Bernoulli, 12(5):863–888, 2006.
  • [23] A.W. van der Vaart and J.H. van Zanten. Rates of contraction of posterior distributions based on Gaussian process priors., Ann. Statist., 36(3) :1435–1463, 2008.
  • [24] A.W. van der Vaart and J.H. van Zanten. Adaptive Bayesian estimation using a Gaussian random field with inverse gamma bandwidth., Ann. Statist., 37(5B) :2655–2675, 2009.
  • [25] A.W. Van der Vaart and J.H. Van Zanten., Reproducing Kernel Hilbert Spaces of Gaussian priors, volume 3 of IMS Collections, pages 200–222. Institute of Mathematical Statistics, 2008.