MAP estimators and their consistency in Bayesian nonparametric inverse problems

Inverse Problems

Volumn 29, Issue 9, 2013, Pages

  Dashti, M ^a   Law, K J H ^b   Stuart, A M ^c   Voss, J ^d  

^a UNIVERSITY OF SUSSEX   (United Kingdom)

^b KING ABDULLAH UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Saudi Arabia)

^c UNIVERSITY OF WARWICK   (United Kingdom)

^d UNIVERSITY OF LEEDS   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BAYESIAN APPROACHES; CALCULUS OF VARIATIONS; DISTRIBUTED NOISE; GAUSSIAN RANDOM FIELDS; MAP ESTIMATOR; MAXIMUM A POSTERIORI; NOISY MEASUREMENTS; NON-PARAMETRIC;

BAYESIAN NETWORKS; GAUSSIAN DISTRIBUTION; INVERSE PROBLEMS; MOLECULAR DYNAMICS; NAVIER STOKES EQUATIONS; WEATHER FORECASTING;

ESTIMATION;

EID: 84884126479     PISSN: 02665611     EISSN: 13616420     Source Type: Journal    
DOI: 10.1088/0266-5611/29/9/095017     Document Type: Article

References (35)

1. Agapiou S, Larsson S and Stuart A M 2013 Posterior consistency of the Bayesian approach to linear ill-posed inverse problems Stoc. Proc. App. pp 3828-60
2. Bennett A F 2002 Inverse Modeling of the Ocean and Atmosphere (Cambridge: Cambridge University Press)
3. Bissantz N, Hohage T and Munk A 2004 Consistency and rates of convergence of nonlinear Tikhonov regularization with random noise Inverse Problems 20 1773-89 (Pubitemid 40002902)
4. Bochkina N A 2012 Consistency of the posterior distribution in generalised linear inverse problems arXiv:1211.3382
5. Bogachev V I 1998 Gaussian Measures (Mathematical Surveys and Monographs vol 62) (Providence, RI: American Mathematical Society)
6. Brown L D and Low M G 1996 Asymptotic equivalence of nonparametric regression and white noise Ann. Stat. 24 2384-98
7. Cotter S L, Dashti M, Robinson J C and Stuart A M 2009 Bayesian inverse problems for functions and applications to fluid mechanics Inverse Problems 25 115008
8. Cox S M and Matthews P C 2002 Exponential time differencing for stiff systems J. Comput. Phys. 176 430-55
9. Dembo A and Zeitouni O 1991 Onsager-Machlup functionals and maximum a posteriori estimation for a class of non-Gaussian random fields J. Mutivariate Anal. 36 243-62
10. Da Prato G and Zabczyk J 1992 Stochastic Equations in Infinite Dimensions (Encyclopedia of Mathematics and Its Applications vol 44) (Cambridge: Cambridge University Press)
11. Dürr D and Bach A 1978 The Onsager-Machlup function as Lagrangian for the most probable path of a diffusion process Commun. Math. Phys. 160 153-70
12. Engl H W, Hanke M and Neubauer A 1996 Regularization of Inverse Problems (Mathematics and Its Applications vol 375) (Dordrecht: Kluwer)
13. Fletcher R 2000 Practical Methods of Optimization 2nd edn (New York: Wiley-Blackwell)
14. Galassi M, Davies J, Theiler J, Gough B, Jungman G, Alken P, Booth M and Rossi F 2009 GNU Scientific Library - Reference Manual 3rd edn (Godalming: Network Theory)
15. Ghosal S, Ghosh J K and Ramamoorthi R V 1999 Consistency issues in Bayesian nonparametrics Asymptotics, Nonparameters and Time Series vol 158 (New York: Marcel Dekker Inc) pp 639-68
16. Griebel M and Hegland M A 2010 Finite element method for density estimation with Gaussian process priors SIAM J. Numer. Anal. 47 4759-92
17. Hairer M, Stuart A M and Voss J 2007 Analysis of SPDEs arising in path sampling: II. The nonlinear case Ann. Appl. Probab. 17 1657-706
18. Hairer M, Stuart A M and Voss J 2011 Signal processing problems on function space: Bayesian formulation, stochastic PDEs and effective MCMC methods The Oxford Handbook of Nonlinear Filtering ed D Crisan and B Rozovsky (Oxford: Oxford University Press)
19. Hegland M 2007 Approximate maximum a posteriori with Gaussian process priors Constructive Approximation 26 205-24
20. Hesthaven J S, Gottlieb S and Gottlieb D 2007 Spectral Methods for Time-Dependent Problems (Cambridge Monographs on Applied and Computational Mathematics vol 21) (Cambridge: Cambridge University Press)
21. Ikeda N and Watanabe S 1989 Stochastic Differential Equations and Diffusion Processes 2nd edn (Amsterdam: North-Holland)
22. Kaipio J P and Somersalo E 2005 Statistical and Computational Inverse Problems (Berlin: Springer)
23. Knapik B T, van der Vaart A W and van Zanten J H 2011 Bayesian inverse problems with Gaussian priors Ann. Stat. 39 2626-57
24. Ledoux M and Talagrand M 1991 Probability in Banach Spaces: Isoperimetry and Processes (Berlin: Springer)
25. Lifshits M A 1995 Gaussian Random Functions (Dordrecht: Kluwer)
26. Majda A and Wang X 2006 Non-Linear Dynamics and Statistical Theories for Basic Geophysical Flows (Cambridge: Cambridge University Press)
27. Markussen B 2009 Laplace approximation of transition densities posed as Brownian expectations Stochastic Process. Appl. 119 208-31
28. Mortensen R E 1968 Maximum-likelihood recursive filtering J. Optim. Theory Appl. 2 386-93
29. Ray K 2012 Bayesian inverse problems with non-conjugate priors arXiv:1209.6156
30. Rasmussen C E and Williams C K I 2006 Gaussian Processes for Machine Learning (Cambridge, MA: MIT Press)
31. Robinson J C 2001 Infinite-Dimensional Dynamical Systems (Cambridge Texts in Applied Mathematics) (Cambridge: Cambridge University Press)
32. Rogers C 2010 Least-action filtering http://www.statslab.cam.ac.uk/ ∼chris/laf.pdf
33. Stuart A M 2010 Inverse problems: a Bayesian approach Acta Numer. 19 451-559
34. 2 curves Ann. Probab. 17 1037-54
35. Zeitouni O 2000 MAP estimation of diffusions: an updated account Systems Theory: Modeling, Analysis and Control (Kluwer International Series in Engineering and Computer Science vol 518) p 145