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SUMMARY:Bayesian inference and uncertainty quantification in non-linear in
 verse problems with Gaussian priors - Katerina Papagiannouli (Max Planck I
 nstitute for Mathematics in the Sciences)
DTSTART:20250603T143000Z
DTEND:20250603T153000Z
UID:TALK230824@talks.cam.ac.uk
DESCRIPTION:We study asymptotic frequentist coverage and approximately Gau
 ssian properties of&nbsp\;Bayes posterior credible sets in nonlinear inver
 se problems when a Gaussian prior is placed on the parameter of the PDE. T
 he aim is to ensure valid frequentist coverage of Bayes credible intervals
  when estimating continuous linear functionals of the parameter. Our resul
 ts show that Bayes credible intervals have conservative coverage under cer
 tain smoothness assumptions on the parameter and a compatibility condition
  between the likelihood and the prior\, regardless of whether an efficient
  limit exists or Bernstein von-Mises (BvM) theorem holds. In the latter ca
 se\, our work yields a result with more relaxed sufficient conditions than
  previous works. We illustrate the practical utility of the results throug
 h the example of estimating the conductivity coefficient of a second order
  elliptic PDE\, where a near-$N^{-1/2}$ contraction rate and conservative 
 coverage results are obtained or linear functionals that were shown not to
  be estimable efficiently. Bayesian methods are attractive for uncertainty
  quantification but assume knowledge of the likelihood model or data gener
 ation process. This assumption is difficult to justify in many inverse pro
 blems. We study a contraction rate of posterior distributions if the model
  is misspecified. Given a prior distribution and a random sample from a di
 stribution $P_0$\, which may not be in the support of the prior\, we show 
 that the posterior concentrates its mass near the points in the support of
  the prior that minimize the Kullback-Leibler divergence with respect to $
 P_0$. Convexity is not required and the existence of a minimizer is not ta
 ken for granted. Joint work with Y. Baek and S. Mukherjee
LOCATION:Seminar Room 1\, Newton Institute
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