  Uncertainty quantification has received increasing attention in machine
learning in the recent past. In particular, a distinction between aleatoric and
epistemic uncertainty has been found useful in this regard. The latter refers
to the learner's (lack of) knowledge and appears to be especially difficult to
measure and quantify. In this paper, we analyse a recent proposal based on the
idea of a second-order learner, which yields predictions in the form of
distributions over probability distributions. While standard (first-order)
learners can be trained to predict accurate probabilities, namely by minimising
suitable loss functions on sample data, we show that loss minimisation does not
work for second-order predictors: The loss functions proposed for inducing such
predictors do not incentivise the learner to represent its epistemic
uncertainty in a faithful way.
