Rates of contraction of posterior distributions based on Gaussian process priors
Tóm tắt
Từ khóa
Tài liệu tham khảo
[1] Albert, J. H. and Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. <i>J. Amer. Statist. Assoc.</i> <b>88</b> 422, 669–679.
[2] Ayache, A. and Taqqu, M. S. (2003). Rate optimality of wavelet series approximations of fractional Brownian motion. <i>J. Fourier Anal. Appl.</i> <b>9</b> 451–471.
[3] Borell, C. (1975). The Brunn–Minkowski inequality in Gauss space. <i>Invent. Math.</i> <b>30</b> 207–216.
[5] Choudhuri, N., Ghosal, S. and Roy, A. (2007). Nonparametric binary regression using a Gaussian process prior. <i>Statist. Methodol.</i> <b>4</b> 227–243.
[6] Dzhaparidze, K. and van Zanten, H. (2004). A series expansion of fractional Brownian motion. <i>Probab. Theory Related Fields</i> <b>130</b> 39–55.
[7] Dzhaparidze, K. and van Zanten, H. (2005). Krein’s spectral theory and the Paley–Wiener expansion for fractional Brownian motion. <i>Ann. Probab.</i> <b>33</b> 620–644.
[8] Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. <i>Ann. Statist.</i> <b>28</b> 500–531.
[9] Ghosal, S. and Roy, A. (2006). Posterior consistency in nonparametric regression problem under Gaussian process prior. <i>Ann. Statist.</i> <b>34</b> 2413–2429.
[10] Ghosal, S. and van der Vaart, A. (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities. <i>Ann. Statist.</i> <b>35</b> 697–723.
[11] Ghosal, S. and van der Vaart, A. (2007). Convergence rates of posterior distributions for noniid observations. <i>Ann. Statist.</i> <b>35</b> 192–223.
[13] Hult, H. (2003). Approximating some Volterra type stochastic integrals with applications to parameter estimation. <i>Stochastic Process. Appl.</i> <b>105</b> 1–32.
[14] Iglói, E. (2005). A rate-optimal trigonometric series expansion of the fractional Brownian motion. <i>Electron. J. Probab.</i> <b>10</b> 1381–1397 (electronic).
[15] Kimeldorf, G. S. and Wahba, G. (1970). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. <i>Ann. Math. Statist.</i> <b>41</b> 495–502.
[16] Kuelbs, J., Li, W. V. and Linde, W. (1994). The Gaussian measure of shifted balls. <i>Probab. Theory Related Fields</i> <b>98</b> 143–162.
[17] Kühn, T. and Linde, W. (2002). Optimal series representation of fractional Brownian sheets. <i>Bernoulli</i> <b>8</b> 669–696.
[19] Lenk, P. J. (1988). The logistic normal distribution for Bayesian, nonparametric, predictive densities. <i>J. Amer. Statist. Assoc.</i> <b>83</b> 509–516.
[20] Lenk, P. J. (1991). Towards a practicable Bayesian nonparametric density estimator. <i>Biometrika</i> <b>78</b> 531–543.
[21] Lenk, P. J. (1999). Bayesian inference for semiparametric regression using a Fourier representation. <i>J. R. Stat. Soc. Ser. B Stat. Methodol.</i> <b>61</b> 863–879.
[22] Leonard, T. (1978). Density estimation, stochastic processes and prior information (with discussion). <i>J. Roy. Statist. Soc. Ser. B</i> <b>40</b> 113–146.
[23] Li, W. V. and Linde, W. (1998). Existence of small ball constants for fractional Brownian motions. <i>C. R. Acad. Sci. Paris Sér. I Math.</i> <b>326</b> 1329–1334.
[24] Li, W. V. and Shao, Q.-M. (2001). Gaussian processes: Inequalities, small ball probabilities and applications. In <i>Stochastic Processes</i>: <i>Theory and Methods. Handbook of Statist.</i> <b>19</b> 533–597. North-Holland, Amsterdam.
[25] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. <i>SIAM Rev.</i> <b>10</b> 422–437.
[26] Neal, R. (1996). <i>Bayesian Learning for Neural Networks. Lecture Notes in Statist.</i> <b>118</b>. Springer, New York.
[29] Tokdar, S. and Ghosh, J. (2005). Posterior consistency of Gaussian process priors in density estimation. <i>J. Statist. Plann. Inference</i> <b>137</b> 34–42.
[33] Wahba, G. (1978). Improper priors, spline smoothing and the problem of guarding against model errors in regression. <i>J. Roy. Statist. Soc. Ser. B</i> <b>40</b> 364–372.
[34] Wood, S. and Kohn, R. (1998). A Bayesian approach to robust nonparametric binary regression. <i>J. Amer. Statist. Assoc.</i> <b>93</b> 203–213.
[4] Choi, T. and Schervish, M. (2004). Posterior consistency in nonparametric regression problem under Gaussian process prior. Preprint.
[12] Ghosh, J. K. and Ramamoorthi, R. V. (2003). <i>Bayesian Nonparametrics</i>. Springer, New York.
[18] Ledoux, M. and Talagrand, M. (1991). <i>Probability in Banach Spaces</i>. Springer, Berlin.
[27] Rasmussen, C. E. and Williams, C. K. (2006). <i>Gaussian Processes for Machine Learning</i>. MIT Press, Cambridge, MA.
[28] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). <i>Fractional Integrals and Derivatives</i>. Gordon and Breach, Yverdon.
[30] van der Vaart, A. and Van Zanten, J. (2007). Reproducing kernel Hilbert spaces of Gaussian priors. In <i>Festschrift for J. K. Ghosh. IMS Lecture Note Series</i>, <i>2008</i>. To appear.
[31] van der Vaart, A. W. (1988). <i>Statistical Estimation in Large Parameter Spaces</i>. Math. Centrum, Amsterdam.