  Numerous applications of machine learning involve representing probability
distributions over high-dimensional data. We propose autoregressive quantile
flows, a flexible class of normalizing flow models trained using a novel
objective based on proper scoring rules. Our objective does not require
calculating computationally expensive determinants of Jacobians during training
and supports new types of neural architectures, such as neural autoregressive
flows from which it is easy to sample.
  We leverage these models in quantile flow regression, an approach that
parameterizes predictive conditional distributions with flows, resulting in
improved probabilistic predictions on tasks such as time series forecasting and
object detection. Our novel objective functions and neural flow
parameterizations also yield improvements on popular generation and density
estimation tasks, and represent a step beyond maximum likelihood learning of
flows.
