Unifying supervised learning and VAEs -- automating statistical inference in (astro-)particle physics with amortized conditional normalizing flows

A KL-divergence objective of the joint distribution of data and labels allows to unify supervised learning and variational autoencoders (VAEs) under one umbrella of stochastic variational inference. The unification motivates an extended supervised scheme which allows to calculate a goodness-of-fit p-value for the neural network model. Conditional normalizing flows amortized with a neural network are crucial in this construction. We discuss how they allow to rigorously define coverage for posteriors defined jointly on a product space, e.g. $\mathbb{R}^n \times \mathcal{S}^m$, which encompasses posteriors over directions. Finally, systematic uncertainties are naturally included in the variational viewpoint. In classical likelihood approaches or other machine learning models, the ingredients of (1) systematics, (2) coverage and (3) goodness-of-fit are typically not all available or at least one of them strongly constrained. In contrast, the proposed extended supervised training with amortized normalizing flows accommodates all three of them for variational inference of arbitrary statistical distributions defined on product spaces like $\mathbb{R}^n \times \ldots \times \mathcal{S}^m$ and no fundamental barrier in terms of complexity of the underlying data. It therefore has great potential for the statistical toolbox of the contemporary (astro-)particle physicist.