Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency
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The SMAI journal of computational mathematics, Volume 6 (2020) , pp. 69-100.

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.

Published online: 2020-04-24
DOI: https://doi.org/10.5802/smai-jcm.62
Classification: 62G05,  62C10,  62G20,  45Q05
Keywords: Bayesian inverse problems, hierarchical Bayesian, MAP estimation, optimization, nonparametric inference, hyperparameter inference, consistency of estimators.
@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},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {6},
     year = {2020},
     pages = {69-100},
     doi = {10.5802/smai-jcm.62},
     language = {en},
     url={smai-jcm.centre-mersenne.org/item/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/item/SMAI-JCM_2020__6__69_0/

[1] Sergios Agapiou; Johnathan M. Bardsley; Omiros Papaspiliopoulos; Andrew M. Stuart Analysis of the Gibbs sampler for hierarchical inverse problems, SIAM/ASA J. Uncertain. Quantif., Volume 2 (2014) no. 1, pp. 511-544 | Article | MR 3283919 | Zbl 1308.62097

[2] Sergios Agapiou; Martin Burger; Masoumeh Dashti; Tapio Helin Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems, Inverse Probl., Volume 34 (2018) no. 4, 045002 | | MR 3774703 | Zbl 06866427

[3] Sergios Agapiou; Masoumeh Dashti; Tapio Helin Rates of contraction of posterior distributions based on p-exponential priors (2018) (https://arxiv.org/abs/1811.12244)

[4] Sergios Agapiou; Stig Larsson; Andrew M. Stuart Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems, Stochastic Processes Appl., Volume 123 (2013) no. 10, pp. 3828-3860 | Article | MR 3084161 | Zbl 1284.62289

[5] Sergios Agapiou; Peter Mathé Posterior Contraction in Bayesian Inverse Problems Under Gaussian Priors, New Trends in Parameter Identification for Mathematical Models, Springer, 2018, pp. 1-29 | Zbl 1405.62051

[6] Sergios Agapiou; Andrew M. Stuart; Yuan-Xiang Zhang 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 3215928 | Zbl 1288.62036

[7] James O. Berger Statistical Decision Theory and Bayesian Analysis, Springer, 2013

[8] Alexandros Beskos; Ajay Jasra; Ege A. Muzaffer; Andrew M. Stuart Sequential Monte Carlo methods for Bayesian elliptic inverse problems, Stat. Comput., Volume 25 (2015) no. 4, pp. 727-737 | Article | MR 3360488 | Zbl 1331.65012

[9] Alexandros Beskos; Gareth Roberts; Andrew M. Stuart; Jochen Voss MCMC methods for diffusion bridges, Stoch. Dyn., Volume 8 (2008) no. 03, pp. 319-350 | Article | MR 2444507 | Zbl 1159.65007

[10] Neil K. Chada; Marco A. Iglesias; Lassi Roininen; Andrew M. Stuart Parameterizations for ensemble Kalman inversion, Inverse Probl., Volume 34 (2018) no. 5, 055009 | | MR 3788159 | Zbl 06897207

[11] Victor Chen; Matthew M. Dunlop; Omiros Papaspiliopoulos; Andrew M. Stuart Dimension-Robust MCMC in Bayesian Inverse Problems (2018) (https://arxiv.org/abs/1806.00519)

[12] Christian Clason; Tapio Helin; Remo Kretschmann; Petteri Piiroinen Generalized modes in Bayesian inverse problems, SIAM/ASA J. Uncertain. Quantif., Volume 7 (2019) no. 2, pp. 652-684 | Article | MR 3968262 | Zbl 1430.62053

[13] Simon L. Cotter; Gareth Roberts; Andrew M. Stuart; David White MCMC methods for functions: modifying old algorithms to make them faster, Stat. Sci., Volume 28 (2013) no. 3, pp. 424-446 | Article | MR 3135540 | Zbl 1331.62132

[14] Yair Daon; Georg Stadler Mitigating the Influence of the Boundary on PDE-based Covariance Operators, Inverse Probl. Imaging, Volume 12 (2018) no. 5, pp. 1083-1102 | MR 3836580 | Zbl 1401.65043

[15] Masoumeh Dashti; Kody JH Law; Andrew M. Stuart; Jochen Voss MAP estimators and their consistency in Bayesian nonparametric inverse problems, Inverse Probl., Volume 29 (2013) no. 9, 095017 | | MR 3104933

[16] Masoumeh Dashti; Andrew M. Stuart The Bayesian approach to inverse problems (2017), pp. 311-428

[17] Matthew M. Dunlop; Marco A. Iglesias; Andrew M. Stuart Hierarchical Bayesian level set inversion, Stat. Comput. (2016), pp. 1-30 | Zbl 1384.62084

[18] Joel N. Franklin Well-posed stochastic extensions of ill-posed linear problems, J. Math. Anal. Appl., Volume 31 (1970) no. 3, pp. 682-716 | Article | MR 267654 | Zbl 0198.20601

[19] Arnaud Gloter; Marc Hoffmann Estimation of the Hurst parameter from discrete noisy data, Ann. Stat., Volume 35 (2007) no. 5, pp. 1947-1974 | Article | MR 2363959 | Zbl 1126.62073

[20] Shota Gugushvili; Aad W. van der Vaart; Dong Yan Bayesian inverse problems with partial observations, Trans. A. Razmadze Math. Inst., Volume 172 (2018) no. 3, pp. 388-403 | Article | MR 3870056 | Zbl 1422.62139

[21] Shota Gugushvili; Aad W. van der Vaart; Dong Yan Bayesian linear inverse problems in regularity scales (2018) (https://arxiv.org/abs/1802.08992)

[22] Tapio Helin; Martin Burger Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems, Inverse Probl., Volume 31 (2015) no. 8, 085009 | | MR 3377106 | Zbl 1325.62058

[23] Tapio Helin; Matti Lassas Hierarchical models in statistical inverse problems and the Mumford–Shah functional, Inverse Probl., Volume 27 (2010) no. 1, 015008 | | MR 2746411

[24] Jari Kaipio; Erkki Somersalo Statistical and Computational Inverse Problems Volume 160, Springer, 2006 | Zbl 1068.65022

[25] Ustim Khristenko; Laura Scarabosio; Piotr Swierczynski; Elisabeth Ullmann; Barbara Wohlmuth 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 | Article | Zbl 07118418

[26] Bartek T Knapik; Botond T. Szabó; Aad W. van der Vaart; J. Harry van Zanten 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 | Article | MR 3477780 | Zbl 1334.62039

[27] Bartek T Knapik; Aad W. van der Vaart; J. Harry van Zanten Bayesian recovery of the initial condition for the heat equation, Commun. Stat., Theory Methods, Volume 42 (2013) no. 7, pp. 1294-1313 | Article | MR 3031282 | Zbl 1347.62057

[28] Bartek T Knapik; Aad W. van der Vaart; J. Harry van Zanten Bayesian inverse problems with Gaussian priors, Ann. Stat., Volume 39 (2011) no. 5, pp. 2626-2657 | Article | MR 2906881 | Zbl 1232.62079

[29] Sari Lasanen Non-Gaussian statistical inverse problems. Part I: Posterior distributions, Inverse Probl. Imaging, Volume 6 (2012) no. 2, pp. 215-266 | Article | MR 2942739 | Zbl 1263.62041

[30] Sari Lasanen Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns., Inverse Probl. Imaging, Volume 6 (2012) no. 2, pp. 267-287 | MR 2942740 | Zbl 1263.62042

[31] Markku S. Lehtinen; Lassi Paivarinta; Erkki Somersalo Linear inverse problems for generalised random variables, Inverse Probl., Volume 5 (1989) no. 4, pp. 599-612 | MR 1009041 | Zbl 0681.60015

[32] Finn Lindgren; Håvard Rue; Johan Lindström 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 | Article | MR 2853727 | Zbl 1274.62360

[33] Kevin P Murphy Machine Learning: A Probabilistic Perspective, MIT Press, 2012 | Zbl 1295.68003

[34] Radford M. Neal Bayesian Learning for Neural Networks (1995) (Ph. D. Thesis) | Zbl 0888.62021

[35] Radford M. Neal Monte Carlo implementation of Gaussian process models for Bayesian regression and classification (1997) (https://arxiv.org/abs/physics/9701026)

[36] Richard Nickl Bernstein-von Mises theorems for statistical inverse problems I: Schrödinger equation (2017) (https://arxiv.org/abs/1707.01764)

[37] Richard Nickl; Kolyan Ray Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018) (https://arxiv.org/abs/1810.01702)

[38] Richard Nickl; Jakob Söhl Bernstein-von Mises theorems for statistical inverse problems II: compound Poisson processes, Electron. J. Stat., Volume 13 (2019) no. 2, pp. 3513-3571 | Article | MR 4013745 | Zbl 1429.62168

[39] Richard Nickl; Sara van de Geer; Sven Wang Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems (2018) (https://arxiv.org/abs/1809.08818)

[40] Houman Owhadi; Clint Scovel; Tim Sullivan On the brittleness of Bayesian inference, SIAM Rev., Volume 57 (2015) no. 4, pp. 566-582 | Article | MR 3419871 | Zbl 1341.62094

[41] Omiros Papaspiliopoulos; Gareth Roberts; Martin Sköld A general framework for the parametrization of hierarchical models, Stat. Sci. (2007), pp. 59-73 | Article | MR 2408661 | Zbl 1246.62195

[42] Kolyan Ray Bayesian inverse problems with non-conjugate priors, Electron. J. Stat., Volume 7 (2013), pp. 2516-2549 | MR 3117105 | Zbl 1294.62107

[43] Gareth Roberts; Osnat Stramer On inference for partially observed nonlinear diffusion models using the Metropolis–Hastings algorithm, Biometrika, Volume 88 (2001) no. 3, pp. 603-621 | Article | MR 1859397 | Zbl 0985.62066

[44] Lassi Roininen; Janne M. J. Huttunen; Sari Lasanen Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography, Inverse Probl. Imaging, Volume 8 (2014) no. 2, pp. 561-586 | Article | Zbl 1302.65245

[45] Andrew M. Stuart Inverse problems: a Bayesian perspective, Acta Numerica, Volume 19, Cambridge University Press, 2010, pp. 451-559 | Article | MR 2652785 | Zbl 1242.65142

[46] Aad W. van der Vaart; Jon A. Wellner Weak convergence, Weak convergence and empirical processes, Springer, 1996, pp. 16-28 | Article

[47] J. Harry van Zanten A Note on Consistent Estimation of Multivariate Parameters in Ergodic Diffusion Models, Scand. J. Stat., Volume 28 (2001) no. 4, pp. 617-623 | Article | MR 1876503 | Zbl 1010.62075

[48] Yaming Yu; Xiao-Li Meng 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 2878987