  We combine concept-based neural networks with generative, flow-based
classifiers into a novel, intrinsically explainable, exactly invertible
approach to supervised learning. Prototypical neural networks, a type of
concept-based neural network, represent an exciting way forward in realizing
human-comprehensible machine learning without concept annotations, but a
human-machine semantic gap continues to haunt current approaches. We find that
reliance on indirect interpretation functions for prototypical explanations
imposes a severe limit on prototypes' informative power. From this, we posit
that invertibly learning prototypes as distributions over the latent space
provides more robust, expressive, and interpretable modeling. We propose one
such model, called ProtoFlow, by composing a normalizing flow with Gaussian
mixture models. ProtoFlow (1) sets a new state-of-the-art in joint generative
and predictive modeling and (2) achieves predictive performance comparable to
existing prototypical neural networks while enabling richer interpretation.
