Unifying supervised learning and VAEs -- automating statistical inference in high-energy physics

A KL-divergence objective of the joint distribution of data and labels allows to unify supervised learning, variational autoencoders (VAEs) and semi-supervised learning under one umbrella of variational inference. This viewpoint has several advantages. For VAEs, it clarifies the interpretation of encoder and decoder parts. For supervised learning, it re-iterates that the training procedure approximates the true posterior over labels and can always be viewed as approximate likelihood-free inference. This is typically not discussed, even though the derivation is well-known in the literature. In the context of semi-supervised learning it motivates an extended supervised scheme which allows to calculate a goodness-of-fit p-value using posterior predictive simulations. Flow-based networks with a standard normal base distribution are crucial. We discuss how they allow to rigorously define coverage for arbitrary joint posteriors on $\mathbb{R}^n \times \mathcal{S}^m$, which encompasses posteriors over directions. Finally, systematic uncertainties are naturally included in the variational viewpoint. With the three ingredients of (1) systematics, (2) coverage and (3) goodness-of-fit, flow-based neural networks have the potential to replace a large part of the statistical toolbox of the contemporary high-energy physicist.