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SUMMARY:Bayesian Nonparametric inference for inverse problems using Gaussi
 an priors - Matteo Giordano\, Univeristy of Cambridge
DTSTART:20191106T160000Z
DTEND:20191106T170000Z
UID:TALK134524@talks.cam.ac.uk
CONTACT:Jan Bohr
DESCRIPTION:*Abstract.* In a statistical inverse problem one collect noisy
  indirect measurements of an unknown physical quantity of interest. In the
  last decade\, the Bayesian approach to inference for inverse problems has
  received increasing attention\, mainly due to the fact that a) it can be 
 efficiently implemented in practice using MCMC methods\, and b) it provide
 s a principle framework to perform uncertainty quantification and test sci
 entific hypotheses.\n\nThe talk will present some results on the validatio
 n of Bayesian nonparametric procedures based on standard Gaussian process 
 priors. A general framework is considered\, where the posterior distributi
 ons arising from a flexible class of Gaussian priors is shown to concentra
 te around the ground truth generating the data at optimal rate in “predi
 ction-risk”. An important (nonlinear) example consists in the recovery o
 f the diffusivity in an elliptic PDE in divergence form\, for which the a 
 convergence rate for the posterior mean estimator to the unknown is obtain
 ed. Finally\, in a related linear inverse problems\, a Bernstein-von Mises
  limit is derived\, that entails the convergence of the posterior distribu
 tion to a fixed Gaussian measure whose covariance structure attains the in
 formation lower bound. As a result\, credible sets are shown to have asymp
 totically the correct coverage and to shrink at optimal rate.
LOCATION:MR14\, Centre for Mathematical Sciences
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