TY  - JOUR
A1  - Ayanbayev, Birzhan
A1  - Klebanov, Ilja
A1  - Li, Han Cheng
A1  - Sullivan, Tim J.
T1  - Gamma-convergence of Onsager-Machlup functionals
BT  - I. With applications to maximum a posteriori estimation in Bayesian inverse problems
JF  - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data
N2  - 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.
KW  - Bayesian inverse problems
KW  - Gamma-convergence
KW  - maximum a posteriori
KW  - estimation
KW  - Onsager-Machlup functional
KW  - small ball probabilities;
KW  - transition path theory
Y1  - 2021
U6  - https://doi.org/10.1088/1361-6420/ac3f81
SN  - 0266-5611
SN  - 1361-6420
VL  - 38
IS  - 2
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Ayanbayev, Birzhan
A1  - Klebanov, Ilja
A1  - Lie, Han Cheng
A1  - Sullivan, Tim J.
T1  - Gamma-convergence of Onsager-Machlup functionals
BT  - II. Infinite product measures on Banach spaces
JF  - Inverse problems : an international journal of inverse problems, inverse methods and computerised inversion of data
N2  - 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.
KW  - Bayesian inverse problems
KW  - Gamma-convergence
KW  - maximum a posteriori
KW  - estimation
KW  - Onsager-Machlup functional
KW  - small ball probabilities
KW  - transition path theory
Y1  - 2021
U6  - https://doi.org/10.1088/1361-6420/ac3f82
SN  - 0266-5611
SN  - 1361-6420
VL  - 38
IS  - 2
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Reich, Sebastian
A1  - Weissmann, Simon
T1  - Fokker-Planck particle systems for Bayesian inference: computational approaches
JF  - SIAM ASA journal on uncertainty quantification
N2  - 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.
KW  - Bayesian inverse problems
KW  - Fokker-Planck equation
KW  - gradient flow
KW  - affine
KW  - invariance
KW  - gradient-free sampling methods
KW  - localization
Y1  - 2021
U6  - https://doi.org/10.1137/19M1303162
SN  - 2166-2525
VL  - 9
IS  - 2
SP  - 446
EP  - 482
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -