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Hybrid Samplers for Ill‐Posed Inverse Problems

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  • RADU HERBEI
  • IAN W. McKEAGUE

Abstract

. In the Bayesian approach to ill‐posed inverse problems, regularization is imposed by specifying a prior distribution on the parameters of interest and Markov chain Monte Carlo samplers are used to extract information about its posterior distribution. The aim of this paper is to investigate the convergence properties of the random‐scan random‐walk Metropolis (RSM) algorithm for posterior distributions in ill‐posed inverse problems. We provide an accessible set of sufficient conditions, in terms of the observational model and the prior, to ensure geometric ergodicity of RSM samplers of the posterior distribution. We illustrate how these conditions can be checked in an application to the inversion of oceanographic tracer data.

Suggested Citation

  • RADU HERBEI & IAN W. McKEAGUE, 2009. "Hybrid Samplers for Ill‐Posed Inverse Problems," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 839-853, December.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:4:p:839-853
    DOI: 10.1111/j.1467-9469.2009.00649.x
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    References listed on IDEAS

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    1. Jarner, Søren Fiig & Hansen, Ernst, 2000. "Geometric ergodicity of Metropolis algorithms," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 341-361, February.
    2. Axel Gandy & Uwe Jensen, 2005. "On Goodness‐of‐Fit Tests for Aalen's Additive Risk Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(3), pages 425-445, September.
    3. H. Haario & M. Laine & M. Lehtinen & E. Saksman & J. Tamminen, 2004. "Markov chain Monte Carlo methods for high dimensional inversion in remote sensing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 591-607, August.
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    1. Radu Herbei & L. Mark Berliner, 2014. "Estimating Ocean Circulation: An MCMC Approach With Approximated Likelihoods via the Bernoulli Factory," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 944-954, September.

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