# American Institute of Mathematical Sciences

May  2012, 6(2): 267-287. doi: 10.3934/ipi.2012.6.267

## Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns

 1 Department of Mathematical Sciences, P.O. Box 3000, 90014 University of Oulu

Received  June 2009 Revised  January 2012 Published  May 2012

In practical statistical inverse problems, one often considers only finite-dimensional unknowns and investigates numerically their posterior probabilities. As many unknowns are function-valued, it is of interest to know whether the estimated probabilities converge when the finite-dimensional approximations of the unknown are refined. In this work, the generalized Bayes formula is shown to be a powerful tool in the convergence studies. With the help of the generalized Bayes formula, the question of convergence of the posterior distributions is returned to the convergence of the finite-dimensional (or any other) approximations of the unknown. The approach allows many prior distributions while the restrictions are mainly for the noise model and the direct theory. Three modes of convergence of posterior distributions are considered -- weak convergence, setwise convergence and convergence in variation. The convergence of conditional mean estimates is also studied.
Citation: Sari Lasanen. Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns. Inverse Problems & Imaging, 2012, 6 (2) : 267-287. doi: 10.3934/ipi.2012.6.267
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