Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators

Hanne Kekkonen*, Matti Lassas, Samuli Siltanen

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

14 Citations (Scopus)


The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable U (x, w) is Mδ (y, w) = A (U (x, w)) + δ>(y, w), where A is a finitely many orders smoothing linear hypoelliptic operator and δ> 0 is the noise magnitude. The covariance operator CU of U is smoothing of order 2r, self-adjoint, injective and elliptic pseudodifferential operator. If ϵ was taking values in L2 then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem Tδ (mδ) = arg min μϵHr{ Au-mδ;2L 2+δ2CU -1/2u2L 2} However, Gaussian white noise does not take values in L2but in H-s where s>0 is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of the conditional mean estimate to the correct solution as δ → 0 is proven in appropriate function spaces using microlocal analysis. Also the frequentist posterior contractions rates are studied.

Original languageEnglish
Article number085005
JournalInverse Problems
Issue number8
Publication statusPublished - 23 Jun 2016
Externally publishedYes


  • Bayesian inverse problem
  • convergence rate
  • microlocal analysis
  • posterior consistency
  • white noise

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