  The Bayesian approach to solving inverse problems relies on the choice of a
prior. This critical ingredient allows the formulation of expert knowledge or
physical constraints in a probabilistic fashion and plays an important role for
the success of the inference. Recently, Bayesian inverse problems were solved
using generative models as highly informative priors. Generative models are a
popular tool in machine learning to generate data whose properties closely
resemble those of a given database. Typically, the generated distribution of
data is embedded in a low-dimensional manifold. For the inverse problem, a
generative model is trained on a database that reflects the properties of the
sought solution, such as typical structures of the tissue in the human brain in
magnetic resonance (MR) imaging. The inference is carried out in the
low-dimensional manifold determined by the generative model which strongly
reduces the dimensionality of the inverse problem. However, this proceeding
produces a posterior that admits no Lebesgue density in the actual variables
and the accuracy reached can strongly depend on the quality of the generative
model. For linear Gaussian models we explore an alternative Bayesian inference
based on probabilistic generative models which is carried out in the original
high-dimensional space. A Laplace approximation is employed to analytically
derive the required prior probability density function induced by the
generative model. Properties of the resulting inference are investigated.
Specifically, we show that derived Bayes estimates are consistent, in contrast
to the approach employing the low-dimensional manifold of the generative model.
The MNIST data set is used to construct numerical experiments which confirm our
theoretical findings.
