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: 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},
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/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] 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] 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] 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 | Article | MR 3084161 | Zbl 1284.62289
[5] 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] 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] 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 | Article | MR 3360488 | Zbl 1331.65012
[9] MCMC methods for diffusion bridges, Stoch. Dyn., Volume 8 (2008) no. 03, pp. 319-350 | Article | MR 2444507 | Zbl 1159.65007
[10] Parameterizations for ensemble Kalman inversion, Inverse Probl., Volume 34 (2018) no. 5, 055009 | MR 3788159 | Zbl 06897207
[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 | Article | MR 3968262 | Zbl 1430.62053
[13] 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] Mitigating the Influence of the Boundary on PDE-based Covariance Operators, Inverse Probl. Imaging, Volume 12 (2018) no. 5, pp. 1083-1102 | Article | MR 3836580 | Zbl 1401.65043
[15] MAP estimators and their consistency in Bayesian nonparametric inverse problems, Inverse Probl., Volume 29 (2013) no. 9, 095017 | MR 3104933
[16] The Bayesian approach to inverse problems (2017), pp. 311-428
[17] Hierarchical Bayesian level set inversion, Stat. Comput. (2016), pp. 1-30 | Zbl 1384.62084
[18] 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] 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] 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] Bayesian linear inverse problems in regularity scales (2018) (https://arxiv.org/abs/1802.08992)
[22] 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] Hierarchical models in statistical inverse problems and the Mumford–Shah functional, Inverse Probl., Volume 27 (2010) no. 1, 015008 | MR 2746411
[24] Statistical and Computational Inverse Problems, 160, Springer, 2006 | Zbl 1068.65022
[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 | Article | Zbl 07118418
[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 | Article | MR 3477780 | Zbl 1334.62039
[27] 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] Bayesian inverse problems with Gaussian priors, Ann. Stat., Volume 39 (2011) no. 5, pp. 2626-2657 | Article | MR 2906881 | Zbl 1232.62079
[29] 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] Non-Gaussian statistical inverse problems. Part II: Posterior convergence for approximated unknowns., Inverse Probl. Imaging, Volume 6 (2012) no. 2, pp. 267-287 | Article | MR 2942740 | Zbl 1263.62042
[31] Linear inverse problems for generalised random variables, Inverse Probl., Volume 5 (1989) no. 4, pp. 599-612 | Article | MR 1009041 | Zbl 0681.60015
[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 | Article | MR 2853727 | Zbl 1274.62360
[33] Machine Learning: A Probabilistic Perspective, MIT Press, 2012 | Zbl 1295.68003
[34] Bayesian Learning for Neural Networks (1995) (Ph. D. Thesis) | Zbl 0888.62021
[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 07227745
[37] Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018) (https://arxiv.org/abs/1810.01702) | Zbl 07241595
[38] 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] Convergence rates for Penalised Least Squares Estimators in PDE-constrained regression problems (2018) (https://arxiv.org/abs/1809.08818) | Zbl 1436.62163
[40] On the brittleness of Bayesian inference, SIAM Rev., Volume 57 (2015) no. 4, pp. 566-582 | Article | MR 3419871 | Zbl 1341.62094
[41] A general framework for the parametrization of hierarchical models, Stat. Sci. (2007), pp. 59-73 | Article | MR 2408661 | Zbl 1246.62195
[42] Bayesian inverse problems with non-conjugate priors, Electron. J. Stat., Volume 7 (2013), pp. 2516-2549 | MR 3117105 | Zbl 1294.62107
[43] 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] 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] Inverse problems: a Bayesian perspective, Acta Numerica, Volume 19, Cambridge University Press, 2010, pp. 451-559 | Article | MR 2652785 | Zbl 1242.65142
[46] Weak convergence, Weak convergence and empirical processes, Springer, 1996, pp. 16-28 | Article | Zbl 0862.60002
[47] 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] 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