Statistics and Computing

, Volume 27, Issue 6, pp 1555–1584 | Cite as

Hierarchical Bayesian level set inversion

  • Matthew M. DunlopEmail author
  • Marco A. Iglesias
  • Andrew M. Stuart


The level set approach has proven widely successful in the study of inverse problems for interfaces, since its systematic development in the 1990s. Recently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However, the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be circumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful consideration of the development of algorithms which encode probability measure equivalences as the hierarchical parameter is varied, this leads to well-defined Gibbs-based MCMC methods found by alternating Metropolis–Hastings updates of the level set function and the hierarchical parameter. These methods demonstrably outperform non-hierarchical Bayesian level set methods.


Inverse problems for interfaces Level set inversion Hierarchical Bayesian methods 



AMS is grateful to DARPA, EPSRC, and ONR for the financial support. MMD was supported by the EPSRC-funded MASDOC graduate training program. The authors are grateful to Dan Simpson for helpful discussions. The authors are also grateful for discussions with Omiros Papaspiliopoulos about links with probit. The authors would also like to thank the two anonymous referees for their comments that have helped improve the quality of the paper. This research utilized Queen Mary’s MidPlus computational facilities, supported by QMUL Research-IT and funded by EPSRC grant EP/K000128/1.

Supplementary material

11222_2016_9704_MOESM1_ESM.pdf (675 kb)
Supplementary material 1 (pdf 675 KB)


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Computing & Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  2. 2.School of Mathematical SciencesUniversity of NottinghamNottinghamUK

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