  Normalising flows (NFs) for discrete data are challenging because
parameterising bijective transformations of discrete variables requires
predicting discrete/integer parameters. Having a neural network architecture
predict discrete parameters takes a non-differentiable activation function (eg,
the step function) which precludes gradient-based learning. To circumvent this
non-differentiability, previous work has employed biased proxy gradients, such
as the straight-through estimator. We present an unbiased alternative where
rather than deterministically parameterising one transformation, we predict a
distribution over latent transformations. With stochastic transformations, the
marginal likelihood of the data is differentiable and gradient-based learning
is possible via score function estimation. To test the viability of
discrete-data NFs we investigate performance on binary MNIST. We observe great
challenges with both deterministic proxy gradients and unbiased score function
estimation. Whereas the former often fails to learn even a shallow
transformation, the variance of the latter could not be sufficiently controlled
to admit deeper NFs.
