The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; it allows for principled learning of hyperparameters; and it can provide uncertainty quantification. The posterior distribution may in principle be sampled by means of MCMC or SMC methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization; it is important to study MAP estimators, however, because they provide a link with classical optimization approaches to inverse problems and the Bayesian link may be used to improve upon classical optimization approaches. The lack of invariance of MAP estimators under change of parameterization is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises naturally when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein–Uhlenbeck process. The applicability of the results is demonstrated concretely with the study of hierarchical Whittle–Matérn and ARD priors.
DOI: 10.5802/smai-jcm.62
Keywords: Bayesian inverse problems, hierarchical Bayesian, MAP estimation, optimization, nonparametric inference, hyperparameter inference, consistency of estimators.
CC-BY-NC-ND 4.0
@article{SMAI-JCM_2020__6__69_0,
author = {Matthew M. Dunlop and Tapio Helin and Andrew M. Stuart},
title = {Hyperparameter {Estimation} in {Bayesian} {MAP} {Estimation:} {Parameterizations} and {Consistency}},
journal = {The SMAI Journal of computational mathematics},
pages = {69--100},
publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
volume = {6},
year = {2020},
doi = {10.5802/smai-jcm.62},
mrnumber = {4100532},
zbl = {07207994},
language = {en},
url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.62/}
}
TY - JOUR AU - Matthew M. Dunlop AU - Tapio Helin AU - Andrew M. Stuart TI - Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency JO - The SMAI Journal of computational mathematics PY - 2020 DA - 2020/// SP - 69 EP - 100 VL - 6 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.62/ UR - https://www.ams.org/mathscinet-getitem?mr=4100532 UR - https://zbmath.org/?q=an%3A07207994 UR - https://doi.org/10.5802/smai-jcm.62 DO - 10.5802/smai-jcm.62 LA - en ID - SMAI-JCM_2020__6__69_0 ER -
%0 Journal Article %A Matthew M. Dunlop %A Tapio Helin %A Andrew M. Stuart %T Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency %J The SMAI Journal of computational mathematics %D 2020 %P 69-100 %V 6 %I Société de Mathématiques Appliquées et Industrielles %U https://doi.org/10.5802/smai-jcm.62 %R 10.5802/smai-jcm.62 %G en %F SMAI-JCM_2020__6__69_0
Matthew M. Dunlop; Tapio Helin; Andrew M. Stuart. Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency. The SMAI Journal of computational mathematics, Volume 6 (2020), pp. 69-100. doi : 10.5802/smai-jcm.62. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.62/
[1] Analysis of the Gibbs sampler for hierarchical inverse problems, SIAM/ASA J. Uncertain. Quantif., Volume 2 (2014) no. 1, pp. 511-544 | DOI | MR | Zbl
[2] Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems, Inverse Probl., Volume 34 (2018) no. 4, 045002 | MR | Zbl
[3] Rates of contraction of posterior distributions based on -exponential priors (2018) (https://arxiv.org/abs/1811.12244)
[4] Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems, Stochastic Processes Appl., Volume 123 (2013) no. 10, pp. 3828-3860 | DOI | MR | Zbl
[5] Posterior Contraction in Bayesian Inverse Problems Under Gaussian Priors, New Trends in Parameter Identification for Mathematical Models, Springer, 2018, pp. 1-29 | Zbl
[6] Bayesian posterior contraction rates for linear severely ill-posed inverse problems, J. Inverse Ill-Posed Probl., Volume 22 (2014) no. 3, pp. 297-321 | MR | Zbl
[7] Statistical Decision Theory and Bayesian Analysis, Springer, 2013
[8] Sequential Monte Carlo methods for Bayesian elliptic inverse problems, Stat. Comput., Volume 25 (2015) no. 4, pp. 727-737 | DOI | MR | Zbl
[9] MCMC methods for diffusion bridges, Stoch. Dyn., Volume 8 (2008) no. 03, pp. 319-350 | DOI | MR | Zbl
[10] Parameterizations for ensemble Kalman inversion, Inverse Probl., Volume 34 (2018) no. 5, 055009 | MR | Zbl
[11] Dimension-Robust MCMC in Bayesian Inverse Problems (2018) (https://arxiv.org/abs/1806.00519)
[12] Generalized modes in Bayesian inverse problems, SIAM/ASA J. Uncertain. Quantif., Volume 7 (2019) no. 2, pp. 652-684 | DOI | MR | Zbl
[13] MCMC methods for functions: modifying old algorithms to make them faster, Stat. Sci., Volume 28 (2013) no. 3, pp. 424-446 | DOI | MR | Zbl
[14] Mitigating the Influence of the Boundary on PDE-based Covariance Operators, Inverse Probl. Imaging, Volume 12 (2018) no. 5, pp. 1083-1102 | DOI | MR | Zbl
[15] MAP estimators and their consistency in Bayesian nonparametric inverse problems, Inverse Probl., Volume 29 (2013) no. 9, 095017 | MR
[16] The Bayesian approach to inverse problems, Springer (2017), pp. 311-428
[17] Hierarchical Bayesian level set inversion, Stat. Comput. (2016), pp. 1-30 | Zbl
[18] Well-posed stochastic extensions of ill-posed linear problems, J. Math. Anal. Appl., Volume 31 (1970) no. 3, pp. 682-716 | DOI | MR | Zbl
[19] et al. Estimation of the Hurst parameter from discrete noisy data, Ann. Stat., Volume 35 (2007) no. 5, pp. 1947-1974 | DOI | MR | Zbl
[20] Bayesian inverse problems with partial observations, Trans. A. Razmadze Math. Inst., Volume 172 (2018) no. 3, pp. 388-403 | DOI | MR | Zbl
[21] Bayesian linear inverse problems in regularity scales (2018) (https://arxiv.org/abs/1802.08992) | Zbl
[22] Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems, Inverse Probl., Volume 31 (2015) no. 8, 085009 | MR | Zbl
[23] Hierarchical models in statistical inverse problems and the Mumford–Shah functional, Inverse Probl., Volume 27 (2010) no. 1, 015008 | MR
[24] Statistical and Computational Inverse Problems, 160, Springer, 2006 | Zbl
[25] Analysis of boundary effects on PDE-based sampling of Whittle-Matérn random fields, SIAM/ASA J. Uncertain. Quantif., Volume 7 (2019) no. 3, pp. 948-974 | DOI | Zbl
[26] Bayes procedures for adaptive inference in inverse problems for the white noise model, Probab. Theory Relat. Fields, Volume 164 (2016) no. 3-4, pp. 771-813 | DOI | MR | Zbl
[27] Bayesian recovery of the initial condition for the heat equation, Commun. Stat., Theory Methods, Volume 42 (2013) no. 7, pp. 1294-1313 | DOI | MR | Zbl
[28] et al. Bayesian inverse problems with Gaussian priors, Ann. Stat., Volume 39 (2011) no. 5, pp. 2626-2657 | DOI | MR | Zbl
[29] Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. Imaging, Volume 6 (2012) no. 2, pp. 215-266 | DOI | MR | Zbl
[30] Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns., Inverse Probl. Imaging, Volume 6 (2012) no. 2, pp. 267-287 | DOI | MR | Zbl
[31] Linear inverse problems for generalised random variables, Inverse Probl., Volume 5 (1989) no. 4, pp. 599-612 | DOI | MR | Zbl
[32] An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc., Ser. B, Stat. Methodol., Volume 73 (2011) no. 4, pp. 423-498 | DOI | MR | Zbl
[33] Machine Learning: A Probabilistic Perspective, MIT Press, 2012 | Zbl
[34] Bayesian Learning for Neural Networks, Ph. D. Thesis, University of Toronto (1995) | Zbl
[35] Monte Carlo implementation of Gaussian process models for Bayesian regression and classification (1997) (https://arxiv.org/abs/physics/9701026)
[36] Bernstein-von Mises theorems for statistical inverse problems I: Schrödinger equation (2017) (https://arxiv.org/abs/1707.01764) | Zbl
[37] Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018) (https://arxiv.org/abs/1810.01702) | Zbl
[38] Bernstein-von Mises theorems for statistical inverse problems II: compound Poisson processes, Electron. J. Stat., Volume 13 (2019) no. 2, pp. 3513-3571 | DOI | MR | Zbl
[39] Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems (2018) (https://arxiv.org/abs/1809.08818) | Zbl
[40] On the brittleness of Bayesian inference, SIAM Rev., Volume 57 (2015) no. 4, pp. 566-582 | DOI | MR | Zbl
[41] A general framework for the parametrization of hierarchical models, Stat. Sci. (2007), pp. 59-73 | DOI | MR | Zbl
[42] Bayesian inverse problems with non-conjugate priors, Electron. J. Stat., Volume 7 (2013), pp. 2516-2549 | MR | Zbl
[43] On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm, Biometrika, Volume 88 (2001) no. 3, pp. 603-621 | DOI | MR | Zbl
[44] Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Probl. Imaging, Volume 8 (2014) no. 2, pp. 561-586 | DOI | Zbl
[45] Inverse problems: a Bayesian perspective, Acta Numerica, Volume 19, Cambridge University Press, 2010, pp. 451-559 | DOI | MR | Zbl
[46] Weak convergence, Weak convergence and empirical processes, Springer, 1996, pp. 16-28 | DOI | Zbl
[47] A Note on Consistent Estimation of Multivariate Parameters in Ergodic Diffusion Models, Scand. J. Stat., Volume 28 (2001) no. 4, pp. 617-623 | DOI | MR | Zbl
[48] To center or not to center: that is not the question – an Ancillarity–Sufficiency Interweaving Strategy (ASIS) for boosting MCMC efficiency, J. Comput. Graph. Stat., Volume 20 (2011) no. 3, pp. 531-570 | MR
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