American Institute of Mathematical Sciences

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November  2009, 3(4): 567-597. doi: 10.3934/ipi.2009.3.567

On infinite-dimensional hierarchical probability models in statistical inverse problems

Tapio Helin 1,

1. 

Department of Mathematics and System Analysis, Helsinki University of Technology, P.O. Box 1100 (Otakaari 1 M), FI-02015 TKK, Finland

Received  March 2009 Revised  August 2009 Published  October 2009

In this article, the solution of a statistical inverse problem $M = AU+$ε by the Bayesian approach is studied where $U$ is a function on the unit circle $\T$, i.e., a periodic signal. The mapping $A$ is a smoothing linear operator and ε a Gaussian noise. The connection to the solution of a finite-dimensional computational model $M_{kn} = A_k U_n + $εk is discussed. Furthermore, a novel hierarchical prior model for obtaining edge-preserving conditional mean estimates is introduced. The convergence of the method with respect to finer discretization is studied and the posterior distribution is shown to converge weakly. Finally, theoretical findings are illustrated by a numerical example with simulated data.
Keywords: statistical inversion, signal processing., Bayesian inversion, discretization invariance, Inverse problem, segmentation.
Mathematics Subject Classification: 60F17, 65C2.
Citation: Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems & Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567
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