  This paper proposes an effective treatment of hyperparameters in the Bayesian
inference of a scalar field from indirect observations. Obtaining the joint
posterior distribution of the field and its hyperparameters is challenging. The
infinite dimensionality of the field requires a finite parametrization that
usually involves hyperparameters to reflect the limited prior knowledge. In the
present work, we consider a Karhunen-Lo{\`e}ve (KL) decomposition for the
random field and hyperparameters to account for the lack of prior knowledge of
its autocovariance function. The hyperparameters must be inferred. To
efficiently sample jointly the KL coordinates of the field and the
autocovariance hyperparameters, we introduce a change of measure to reformulate
the joint posterior distribution into a hierarchical Bayesian form. The
likelihood depends only on the field's coordinates in a fixed KL basis, with a
prior conditioned on the hyperparameters. We exploit this structure to derive
an efficient Markov Chain Monte Carlo (MCMC) sampling scheme based on an
adapted Metropolis-Hasting algorithm. We rely on surrogate models (Polynomial
Chaos expansions) of the forward model predictions to further accelerate the
MCMC sampling. A first application to a transient diffusion problem shows that
our method is consistent with other approaches based on a change of coordinates
(Sraj et al., 2016). A second application to a seismic traveltime tomography
highlights the importance of inferring the hyperparameters.
