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Bayesian inverse problems with non-conjugate priors. (English) Zbl 1294.62107
Summary: We investigate the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform.

MSC:
62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
62G05 Nonparametric estimation
62G10 Nonparametric hypothesis testing
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[1] Agapiou, S., Larsson, S. and Stuart, A. M. (2013). Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems. Stochastic Process. Appl. 123 3828-3860. · Zbl 1284.62289
[2] Arbel, J., Gayraud, G. and Rousseau, J. (2013). Bayesian optimal adaptive estimation using a sieve prior. Scand. J. Stat. 40 549-570. · Zbl 1364.62102
[3] Bissantz, N., Hohage, T., Munk, A. and Ruymgaart, F. (2007). Convergence rates of general regularization methods for statistical inverse problems and applications. SIAM J. Numer. Anal. 45 2610-2636. · Zbl 1234.62062
[4] Borell, C. (1975). The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30 207-216. · Zbl 0292.60004
[5] Butucea, C. and Tsybakov, A. B. (2007). Sharp optimality in density deconvolution with dominating bias. II. Teor. Veroyatn. Primen. 52 336-349. · Zbl 1142.62017
[6] Castillo, I., Kerkyacharian, G. and Picard, D. (2013). Thomas Bayes’ walk on manifolds. Probab. Theory Related Fields. · Zbl 1285.62028
[7] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24 034004, 19. · Zbl 1137.62323
[8] Da Prato, G. (2006). An introduction to infinite-dimensional analysis . Universitext . Springer-Verlag, Berlin. Revised and extended from the 2001 original by Da Prato. · Zbl 1065.46001
[9] Dembo, A., Mayer-Wolf, E. and Zeitouni, O. (1995). Exact behavior of Gaussian seminorms. Statist. Probab. Lett. 23 275-280. · Zbl 0830.60030
[10] Folland, G. B. (1999). Real analysis , second ed. Pure and Applied Mathematics (New York) . John Wiley & Sons Inc., New York. Modern techniques and their applications, A Wiley-Interscience Publication. · Zbl 0924.28001
[11] Ghosal, S., Ghosh, J. K. and van der Vaart, A. W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. · Zbl 1105.62315
[12] Ghosal, S. and van der Vaart, A. (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann. Statist. 35 697-723. · Zbl 1117.62046
[13] Giné, E. and Nickl, R. (2011). Rates on contraction for posterior distributions in \(L^{r}\)-metrics, \(1\leq r\leq\infty\). Ann. Statist. 39 2883-2911. · Zbl 1246.62095
[14] Hoffmann-Jørgensen, J., Shepp, L. A. and Dudley, R. M. (1979). On the lower tail of Gaussian seminorms. Ann. Probab. 7 319-342. · Zbl 0424.60041
[15] Holmes, C. and Denison, G. (1999). Bayesian wavelet analysis with a model complexity prior. In Bayesian Statistics 6: proceedings of the sixth Valencia international meeting 769-776. Clarendon Press, Oxford. · Zbl 0976.62023
[16] Huang, T.-M. (2004). Convergence rates for posterior distributions and adaptive estimation. Ann. Statist. 32 1556-1593. · Zbl 1095.62055
[17] Johnstone, I. M., Kerkyacharian, G., Picard, D. and Raimondo, M. (2004). Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 547-573. · Zbl 1046.62039
[18] Johnstone, I. M. and Silverman, B. W. (1990). Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18 251-280. · Zbl 0699.62043
[19] Knapik, B. T., Szabó, B. T., van der Vaart, A. W. and van Zanten, J. H. (2012). Bayes procedures for adaptive inference in inverse problems for the white noise model. · Zbl 1334.62039
[20] Knapik, B. T., van der Vaart, A. W. and van Zanten, J. H. (2011). Bayesian inverse problems with Gaussian priors. Ann. Statist. 39 2626-2657. · Zbl 1232.62079
[21] Knapik, B. T., van der Vaart, A. W. and van Zanten, J. H. (2013). Bayesian recovery of the initial condition for the heat equation. Comm. Statist. Theory Methods 42 1294-1313. · Zbl 1347.62057
[22] Kuelbs, J. and Li, W. V. (1993). Metric entropy and the small ball problem for Gaussian measures. J. Funct. Anal. 116 133-157. · Zbl 0799.46053
[23] Ledoux, M. (2001). The concentration of measure phenomenon . Mathematical Surveys and Monographs 89 . American Mathematical Society, Providence, RI. · Zbl 0995.60002
[24] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201-231. · Zbl 1209.62060
[25] Meyer, Y. (1992). Wavelets and operators . Cambridge Studies in Advanced Mathematics 37 . Cambridge University Press, Cambridge. Translated from the 1990 French original by D. H. Salinger. · Zbl 0776.42019
[26] Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Statist. 9 249-265.
[27] Pensky, M. and Vidakovic, B. (1999). Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 2033-2053. · Zbl 0962.62030
[28] Rudin, W. (1991). Functional analysis , second ed. International Series in Pure and Applied Mathematics . McGraw-Hill Inc., New York. · Zbl 0867.46001
[29] Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687-714. · Zbl 1041.62022
[30] Sytaya, G. N. (1974). On some asymptotic representations of the Gaussian measure in a Hilbert space. Theory of Stochastic Processes, Ukrainian Academy of Sciences, 2 93-104. · Zbl 0282.60022
[31] van der Vaart, A. W. and van Zanten, J. H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36 1435-1463. · Zbl 1141.60018
[32] van der Vaart, A. W. and van Zanten, J. H. (2008). Reproducing kernel Hilbert spaces of Gaussian priors. In Pushing the limits of contemporary statistics: contributions in honor of Jayanta K. Ghosh . Inst. Math. Stat. Collect. 3 200-222. Inst. Math. Statist., Beachwood, OH. · Zbl 1141.60018
[33] van der Vaart, A. W. and van Zanten, J. H. (2009). Adaptive Bayesian estimation using a Gaussian random field with inverse gamma bandwidth. Ann. Statist. 37 2655-2675. · Zbl 1173.62021
[34] Zhao, L. H. (2000). Bayesian aspects of some nonparametric problems. Ann. Statist. 28 532-552. · Zbl 1010.62025
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