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/articles/10.5802/smai-jcm.62/}
}
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/
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