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The Bayesian formulation of inverse problems is attractive for three primaryreasons: it provides a clear modelling framework; means for uncertaintyquantification; and it allows for principled learning of hyperparameters. Theposterior distribution may be explored by sampling methods, but for manyproblems it is computationally infeasible to do so. In this situation maximum aposteriori (MAP) estimators are often sought. Whilst these are relatively cheapto compute, and have an attractive variational formulation, a key drawback istheir lack of invariance under change of parameterization. This is aparticularly significant issue when hierarchical priors are employed to learnhyperparameters. In this paper we study the effect of the choice ofparameterization on MAP estimators when a conditionally Gaussian hierarchicalprior distribution is employed. Specifically we consider the centredparameterization, the natural parameterization in which the unknown state issolved for directly, and the noncentred parameterization, which works with awhitened Gaussian as the unknown state variable, and arises when consideringdimension-robust MCMC algorithms; MAP estimation is well-defined in thenonparametric setting only for the noncentred parameterization. However, weshow that MAP estimates based on the noncentred parameterization are notconsistent as estimators of hyperparameters; conversely, we show that limits offinite-dimensional centred MAP estimators are consistent as the dimension tendsto infinity. We also consider empirical Bayesian hyperparameter estimation,show consistency of these estimates, and demonstrate that they are more robustwith respect to noise than centred MAP estimates. An underpinning conceptthroughout is that hyperparameters may only be recovered up to measureequivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeckprocess.

Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency