@article{AyanbayevKlebanovLietal.2021,
  author    = {Ayanbayev, Birzhan and Klebanov, Ilja and Li, Han Cheng and Sullivan, Tim J.},
  title     = {Gamma-convergence of Onsager-Machlup functionals},
  series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  volume    = {38},
  journal   = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0266-5611},
  doi       = {10.1088/1361-6420/ac3f81},
  pages     = {32},
  year      = {2021},
  abstract  = {The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a maximum a posteriori (MAP) estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager-Machlup (OM) functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the Gamma-convergence of OM functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.},
  language  = {en}
}
@article{AyanbayevKlebanovLieetal.2021,
  author    = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, Tim J.},
  title     = {Gamma-convergence of Onsager-Machlup functionals},
  series = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  volume    = {38},
  journal   = {Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data},
  number    = {2},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {0266-5611},
  doi       = {10.1088/1361-6420/ac3f82},
  pages     = {35},
  year      = {2021},
  abstract  = {We derive Onsager-Machlup functionals for countable product measures on weighted l(p) subspaces of the sequence space R-N. Each measure in the product is a shifted and scaled copy of a reference probability measure on R that admits a sufficiently regular Lebesgue density. We study the equicoercivity and Gamma-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussian, Cauchy, and Besov measures with summability parameter 1 <= p <= 2. Together with part I of this paper, this provides a basis for analysis of the convergence of maximum a posteriori estimators in Bayesian inverse problems and most likely paths in transition path theory.},
  language  = {en}
}
@article{ReichWeissmann2021,
  author    = {Reich, Sebastian and Weissmann, Simon},
  title     = {Fokker-Planck particle systems for Bayesian inference: computational approaches},
  series = {SIAM ASA journal on uncertainty quantification},
  volume    = {9},
  journal   = {SIAM ASA journal on uncertainty quantification},
  number    = {2},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {2166-2525},
  doi       = {10.1137/19M1303162},
  pages     = {446 -- 482},
  year      = {2021},
  abstract  = {Bayesian inference can be embedded into an appropriately defined dynamics in the space of probability measures. In this paper, we take Brownian motion and its associated Fokker-Planck equation as a starting point for such embeddings and explore several interacting particle approximations. More specifically, we consider both deterministic and stochastic interacting particle systems and combine them with the idea of preconditioning by the empirical covariance matrix. In addition to leading to affine invariant formulations which asymptotically speed up convergence, preconditioning allows for gradient-free implementations in the spirit of the ensemble Kalman filter. While such gradient-free implementations have been demonstrated to work well for posterior measures that are nearly Gaussian, we extend their scope of applicability to multimodal measures by introducing localized gradient-free approximations. Numerical results demonstrate the effectiveness of the considered methodologies.},
  language  = {en}
}