2015, 9(3): 895-915. doi: 10.3934/ipi.2015.9.895

Oracle-type posterior contraction rates in Bayesian inverse problems

1. 

School of Mathematical Sciences, Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University, Shanghai, 200433, China, China

2. 

Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany

Received  April 2014 Revised  June 2015 Published  July 2015

We discuss Bayesian inverse problems in Hilbert spaces. The focus is on a fast concentration of the posterior probability around the unknown true solution as expressed in the concept of posterior contraction rates. This concentration is dominated by a parameter which controls the variance of the prior distribution. Previous results determine posterior contraction rates based on known solution smoothness. Here we show that an oracle-type parameter choice is possible. This is done by relating the posterior contraction rate to the root mean squared estimation error. In addition we show that the tail probability, which usually is bounded by using the Chebyshev inequality, has exponential decay, at least for a priori parameter choices. These results implement the exponential concentration of Gaussian measures in Hilbert spaces.
Citation: Kui Lin, Shuai Lu, Peter Mathé. Oracle-type posterior contraction rates in Bayesian inverse problems. Inverse Problems & Imaging, 2015, 9 (3) : 895-915. doi: 10.3934/ipi.2015.9.895
References:
[1]

S. Agapiou, Aspects of Bayesian Inverse Problems,, Ph.D. thesis, (2013).

[2]

S. Agapiou, S. Larsson and A. M. Stuart, Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems,, Stochastic Process. Appl., 123 (2013), 3828. doi: 10.1016/j.spa.2013.05.001.

[3]

S. Agapiou and P. Mathé, Preconditioning the prior to avoid saturation in Bayesian inverse problems,, submitted, (2014).

[4]

F. Bauer and M. A. Lukas, Comparing parameter choice methods for regularization of ill-posed problems,, Math. Comput. Simulation, 81 (2011), 1795. doi: 10.1016/j.matcom.2011.01.016.

[5]

N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise,, Inverse Problems, 20 (2004), 1773. doi: 10.1088/0266-5611/20/6/005.

[6]

G. Blanchard and P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient regularization,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/11/115011.

[7]

A. Caponnetto and E. De Vito, Optimal rates for the regularized least-squares algorithm,, Found. Comput. Math., 7 (2007), 331. doi: 10.1007/s10208-006-0196-8.

[8]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8.

[9]

B. G. Fitzpatrick, Bayesian analysis in inverse problems,, Inverse Problems, 7 (1991), 675. doi: 10.1088/0266-5611/7/5/003.

[10]

J.-P. Florens and A. Simoni, Regularizing priors for linear inverse problems,, Scand. J. Stat., 39 (2012), 214. doi: 10.1111/j.1467-9469.2011.00784.x.

[11]

P. C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,, Numer. Algorithms, 6 (1994), 1. doi: 10.1007/BF02149761.

[12]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, Applied Mathematical Sciences, (2005).

[13]

B. T. Knapik, A. W. van der Vaart and H. van Zanten, Bayesian inverse problems with Gaussian priors,, Ann. Statist., 39 (2011), 2626. doi: 10.1214/11-AOS920.

[14]

S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions,, Inverse Probl. Imaging, 6 (2012), 215. doi: 10.3934/ipi.2012.6.215.

[15]

S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns,, Inverse Probl. Imaging, 6 (2012), 267. doi: 10.3934/ipi.2012.6.267.

[16]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors,, Inverse Problems and Imaging, 3 (2009), 87. doi: 10.3934/ipi.2009.3.87.

[17]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?,, Inverse Problems, 20 (2004), 1537. doi: 10.1088/0266-5611/20/5/013.

[18]

M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1991). doi: 10.1007/978-3-642-20212-4.

[19]

M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables,, Inverse Problems, 5 (1989), 599. doi: 10.1088/0266-5611/5/4/011.

[20]

S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems. SElected Topics,, Inverse and Ill-posed Problems Series, (2013). doi: 10.1515/9783110286496.

[21]

S. Lu and P. Mathé, Discrepancy based model selection in statistical inverse problems,, J. Complexity, 30 (2014), 290. doi: 10.1016/j.jco.2014.02.002.

[22]

A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space,, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385. doi: 10.1007/BF00533743.

[23]

C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems,, Math. Meth. Statist., 23 (2014), 176. doi: 10.3103/S1066530714030028.

[24]

P. Mathé and B. Hofmann, How general are general source conditions?,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/1/015009.

[25]

P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales,, Inverse Problems, 19 (2003), 789. doi: 10.1088/0266-5611/19/3/319.

[26]

P. Mathé and S. V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data,, Math. Comp., 75 (2006), 1913. doi: 10.1090/S0025-5718-06-01873-4.

[27]

P. Mathé and U. Tautenhahn, Enhancing linear regularization to treat large noise,, J. Inverse Ill-Posed Probl., 19 (2011), 859. doi: 10.1515/JIIP.2011.052.

[28]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numer., 19 (2010), 451. doi: 10.1017/S0962492910000061.

show all references

References:
[1]

S. Agapiou, Aspects of Bayesian Inverse Problems,, Ph.D. thesis, (2013).

[2]

S. Agapiou, S. Larsson and A. M. Stuart, Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems,, Stochastic Process. Appl., 123 (2013), 3828. doi: 10.1016/j.spa.2013.05.001.

[3]

S. Agapiou and P. Mathé, Preconditioning the prior to avoid saturation in Bayesian inverse problems,, submitted, (2014).

[4]

F. Bauer and M. A. Lukas, Comparing parameter choice methods for regularization of ill-posed problems,, Math. Comput. Simulation, 81 (2011), 1795. doi: 10.1016/j.matcom.2011.01.016.

[5]

N. Bissantz, T. Hohage and A. Munk, Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise,, Inverse Problems, 20 (2004), 1773. doi: 10.1088/0266-5611/20/6/005.

[6]

G. Blanchard and P. Mathé, Discrepancy principle for statistical inverse problems with application to conjugate gradient regularization,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/11/115011.

[7]

A. Caponnetto and E. De Vito, Optimal rates for the regularized least-squares algorithm,, Found. Comput. Math., 7 (2007), 331. doi: 10.1007/s10208-006-0196-8.

[8]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems,, Mathematics and its Applications, (1996). doi: 10.1007/978-94-009-1740-8.

[9]

B. G. Fitzpatrick, Bayesian analysis in inverse problems,, Inverse Problems, 7 (1991), 675. doi: 10.1088/0266-5611/7/5/003.

[10]

J.-P. Florens and A. Simoni, Regularizing priors for linear inverse problems,, Scand. J. Stat., 39 (2012), 214. doi: 10.1111/j.1467-9469.2011.00784.x.

[11]

P. C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems,, Numer. Algorithms, 6 (1994), 1. doi: 10.1007/BF02149761.

[12]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems,, Applied Mathematical Sciences, (2005).

[13]

B. T. Knapik, A. W. van der Vaart and H. van Zanten, Bayesian inverse problems with Gaussian priors,, Ann. Statist., 39 (2011), 2626. doi: 10.1214/11-AOS920.

[14]

S. Lasanen, Non-Gaussian statistical inverse problems. Part I: Posterior distributions,, Inverse Probl. Imaging, 6 (2012), 215. doi: 10.3934/ipi.2012.6.215.

[15]

S. Lasanen, Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns,, Inverse Probl. Imaging, 6 (2012), 267. doi: 10.3934/ipi.2012.6.267.

[16]

M. Lassas, E. Saksman and S. Siltanen, Discretization-invariant Bayesian inversion and Besov space priors,, Inverse Problems and Imaging, 3 (2009), 87. doi: 10.3934/ipi.2009.3.87.

[17]

M. Lassas and S. Siltanen, Can one use total variation prior for edge-preserving Bayesian inversion?,, Inverse Problems, 20 (2004), 1537. doi: 10.1088/0266-5611/20/5/013.

[18]

M. Ledoux and M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], (1991). doi: 10.1007/978-3-642-20212-4.

[19]

M. S. Lehtinen, L. Päivärinta and E. Somersalo, Linear inverse problems for generalised random variables,, Inverse Problems, 5 (1989), 599. doi: 10.1088/0266-5611/5/4/011.

[20]

S. Lu and S. V. Pereverzev, Regularization Theory for Ill-Posed Problems. SElected Topics,, Inverse and Ill-posed Problems Series, (2013). doi: 10.1515/9783110286496.

[21]

S. Lu and P. Mathé, Discrepancy based model selection in statistical inverse problems,, J. Complexity, 30 (2014), 290. doi: 10.1016/j.jco.2014.02.002.

[22]

A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space,, Z. Wahrsch. Verw. Gebiete, 65 (1984), 385. doi: 10.1007/BF00533743.

[23]

C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems,, Math. Meth. Statist., 23 (2014), 176. doi: 10.3103/S1066530714030028.

[24]

P. Mathé and B. Hofmann, How general are general source conditions?,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/1/015009.

[25]

P. Mathé and S. V. Pereverzev, Geometry of linear ill-posed problems in variable Hilbert scales,, Inverse Problems, 19 (2003), 789. doi: 10.1088/0266-5611/19/3/319.

[26]

P. Mathé and S. V. Pereverzev, Regularization of some linear ill-posed problems with discretized random noisy data,, Math. Comp., 75 (2006), 1913. doi: 10.1090/S0025-5718-06-01873-4.

[27]

P. Mathé and U. Tautenhahn, Enhancing linear regularization to treat large noise,, J. Inverse Ill-Posed Probl., 19 (2011), 859. doi: 10.1515/JIIP.2011.052.

[28]

A. M. Stuart, Inverse problems: A Bayesian perspective,, Acta Numer., 19 (2010), 451. doi: 10.1017/S0962492910000061.

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