Posterior Contraction in Bayesian Inverse Problems Under Gaussian Priors

Part of the Trends in Mathematics book series (TM)


We study Bayesian inference in statistical linear inverse problems with Gaussian noise and priors in a separable Hilbert space setting. We focus our interest on the posterior contraction rate in the small noise limit, under the frequentist assumption that there exists a fixed data-generating value of the unknown. In this Gaussian-conjugate setting, it is convenient to work with the concept of squared posterior contraction (SPC), which is known to upper bound the posterior contraction rate. We use abstract tools from regularization theory, which enable a unified approach to bounding SPC. We review and re-derive several existing results, and establish minimax contraction rates in cases which have not been considered until now. Existing results suffer from a certain saturation phenomenon, when the data-generating element is too smooth compared to the smoothness inherent in the prior. We show how to overcome this saturation in an empirical Bayesian framework by using a non-centered data-dependent prior.


  1. 1.
    S. Agapiou, S. Larsson, A.M. Stuart, Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems. Stoch. Process. Appl. 123(10), 3828–3860 (2013). MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    S. Agapiou, J.M. Bardsley, O. Papaspiliopoulos, A.M. Stuart, Analysis of the Gibbs sampler for hierarchical inverse problems. SIAM/ASA J. Uncertain. Quantif. 2(1), 511–544 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    S. Agapiou, A.M. Stuart, Y.X. Zhang, Bayesian posterior contraction rates for linear severely ill-posed inverse problems. J. Inverse Ill-Posed Prob. 22(3), 297–321 (2014). MathSciNetMATHGoogle Scholar
  4. 4.
    L. Cavalier, Nonparametric statistical inverse problems. Inverse Prob. 24(3), 034004, 19 (2008).
  5. 5.
    M. Dashti, A.M. Stuart, The Bayesian approach to inverse problems (2013). ArXiv e-printsGoogle Scholar
  6. 6.
    L.T. Ding, P. Mathé, Minimax rates for statistical inverse problems under general source conditions (2017). ArXiv e-prints.
  7. 7.
    H.W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, in Mathematics and its Applications, vol. 375 (Kluwer Academic, Dordrecht, 1996). CrossRefMATHGoogle Scholar
  8. 8.
    S. Ghosal, H.K. Ghosh, A.W. Van Der Vaaart, Convergence rates of posterior distributions. Ann. Stat. 28(2), 500–531 (2000). MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    B. Hofmann, P. Mathé, Analysis of profile functions for general linear regularization methods. SIAM J. Numer. Anal. 45(3), 1122–1141(electronic) (2007).
  10. 10.
    B. Knapik, J.B. Salomond, A general approach to posterior contraction in nonparametric inverse problems. Bernoulli (to appear). arXiv preprint arXiv:1407.0335Google Scholar
  11. 11.
    B.T. Knapik, A.W. van der Vaart, J.H. van Zanten, Bayesian inverse problems with Gaussian priors. Ann. Stat. 39(5), 2626–2657 (2011). MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    B.T. Knapik, A.W. van der Vaart, J.H. van Zanten, Bayesian recovery of the initial condition for the heat equation. Comm. Stat. Theory Methods 42(7), 1294–1313 (2013). MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    B.T. Knapik, B.T. Szabó, A.W. van der Vaart, J.H. van Zanten, Bayes procedures for adaptive inference in inverse problems for the white noise model. Probab. Theory Relat. Fields 164, 1–43 (2015)MathSciNetMATHGoogle Scholar
  14. 14.
    M.S. Lehtinen, L. Päivärinta, E. Somersalo, Linear inverse problems for generalised random variables. Inverse Prob. 5(4), 599–612 (1989). MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    K. Lin, S. Lu, P. Mathé, Oracle-type posterior contraction rates in Bayesian inverse problems. Inverse Prob. Imaging 9(3), 895–915 (2015). MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    A. Mandelbaum, Linear estimators and measurable linear transformations on a Hilbert space. Z. Wahrsch. Verw. Gebiete 65(3), 385–397 (1984). MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    P. Mathé, Saturation of regularization methods for linear ill-posed problems in Hilbert spaces. SIAM J. Numer. Anal. 42(3), 968–973 (electronic) (2004).
  18. 18.
    K. Ray, Bayesian inverse problems with non-conjugate priors. Electron. J. Stat. 7, 2516–2549 (2013). MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    B.T. Szabó, A.W. van der Vaart, J.H. van Zanten, Empirical Bayes scaling of Gaussian priors in the white noise model. Electron. J. Stat. 7, 991–1018 (2013). MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    S.J. Vollmer, Posterior consistency for Bayesian inverse problems through stability and regression results. Inverse Prob. 29(12), 125011 (2013).

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© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus
  2. 2.Weierstraß Institute for Applied Analysis and StochasticsBerlinGermany

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