Unifying supervised learning and VAEs -- coverage, systematics and goodness-of-fit in normalizing-flow based neural network models for astro-particle reconstructions

Neural-network based predictions of event properties in astro-particle physics are getting more and more common. However, in many cases the result is just utilized as a point prediction. Statistical uncertainties and coverage (1), systematic uncertainties (2) or a goodness-of-fit measure (3) are often not calculated. Here we describe a certain choice of training and network architecture that allows to incorporate all these properties into a single network model. We show that 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 learning 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. The proposed extended supervised training with amortized normalizing flows incorporates (1) coverage calculation, (2) systematics and (3) a goodness-of-fit measure in a single machine-learning model. There are no constraints on the shape of the involved distributions (e.g. Gaussianity) for these properties to hold, in fact it works with complex multi-modal distributions defined on product spaces like $\mathbb{R}^n \times \mathcal{S}^m$. We see great potential for exploiting this per-event information in event selections or for fast astronomical alerts which require uncertainty guarantees.