Bayesian inference with rescaled Gaussian process priors

We use rescaled Gaussian processes as prior models for functional parameters in nonparametric statistical models. We show how the rate of contraction of the posterior distributions depends on the scaling factor. In particular, we exhibit rescaled Gaussian process priors yielding posteriors that contract around the true parameter at optimal convergence rates. To derive our results we establish bounds on small deviation probabilities for smooth stationary Gaussian processes.

[1]  A. Kolmogorov,et al.  Entropy and "-capacity of sets in func-tional spaces , 1961 .

[2]  G. Wahba,et al.  A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines , 1970 .

[3]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .

[4]  Tom Leonard.  Density Estimation, Stochastic Processes and Prior Information , 1978 .

[5]  K. Parthasarathy.  Introduction to Probability and Measure , 1979 .

[6]  P. Lenk.  The Logistic Normal Distribution for Bayesian, Nonparametric, Predictive Densities , 1988 .

[7]  Peter J. Lenk,et al.  Towards a practicable Bayesian nonparametric density estimator , 1991 .

[8]  J. Kuelbs,et al.  Metric entropy and the small ball problem for Gaussian measures , 1993 .

[9]  L. Werner.  Existence of small ball constants for fractional Brownian motions , 1998 .

[10]  Sally Wood,et al.  A Bayesian Approach to Robust Binary Nonparametric Regression , 1998 .

[11]  Werner Linde,et al.  Approximation, metric entropy and small ball estimates for Gaussian measures , 1999 .

[12]  A. V. D. Vaart,et al.  Convergence rates of posterior distributions , 2000 .

[13]  L. Wasserman,et al.  Rates of convergence of posterior distributions , 2001 .

[14]  Tong Zhang.  From ɛ-entropy to KL-entropy: Analysis of minimum information complexity density estimation , 2006, math/0702653.

[15]  S. Ghosal,et al.  Posterior consistency of Gaussian process prior for nonparametric binary regression , 2006, math/0702686.

[16]  Van Der Vaart,et al.  Convergence rates of posterior distributions for Brownian semimartingale models , 2006 .

[17]  S. Ghosal,et al.  Nonparametric binary regression using a Gaussian process prior , 2007 .

[18]  A. V. D. Vaart,et al.  Convergence rates of posterior distributions for non-i.i.d. observations , 2007, 0708.0491.

[19]  J. Ghosh,et al.  Posterior consistency of logistic Gaussian process priors in density estimation , 2007 .

[20]  S. Walker,et al.  On rates of convergence for posterior distributions in infinite-dimensional models , 2007, 0708.1892.

[21]  A. W. Vaart,et al.  Reproducing kernel Hilbert spaces of Gaussian priors , 2008, 0805.3252.

[22]  Van Der Vaart,et al.  Rates of contraction of posterior distributions based on Gaussian process priors , 2008 .

[23]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.