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

Encoding prior knowledge in the structure of the likelihood