  The inference of deep hierarchical models is problematic due to strong
dependencies between the hierarchies. We investigate a specific transformation
of the model parameters based on the multivariate distributional transform.
This transformation is a special form of the reparametrization trick, flattens
the hierarchy and leads to a standard Gaussian prior on all resulting
parameters. The transformation also transfers all the prior information into
the structure of the likelihood, hereby decoupling the transformed parameters a
priori from each other. A variational Gaussian approximation in this
standardized space will be excellent in situations of relatively uninformative
data. Additionally, the curvature of the log-posterior is well-conditioned in
directions that are weakly constrained by the data, allowing for fast inference
in such a scenario. In an example we perform the transformation explicitly for
Gaussian process regression with a priori unknown correlation structure. Deep
models are inferred rapidly in highly and slowly in poorly informed situations.
The flat model show exactly the opposite performance pattern. A synthesis of
both, the deep and the flat perspective, provides their combined advantages and
overcomes the individual limitations, leading to a faster inference.
