  Aleatoric uncertainty captures the inherent randomness of the data, such as
measurement noise. In Bayesian regression, we often use a Gaussian observation
model, where we control the level of aleatoric uncertainty with a noise
variance parameter. By contrast, for Bayesian classification we use a
categorical distribution with no mechanism to represent our beliefs about
aleatoric uncertainty. Our work shows that explicitly accounting for aleatoric
uncertainty significantly improves the performance of Bayesian neural networks.
We note that many standard benchmarks, such as CIFAR, have essentially no
aleatoric uncertainty. Moreover, we show data augmentation in approximate
inference has the effect of softening the likelihood, leading to
underconfidence and profoundly misrepresenting our honest beliefs about
aleatoric uncertainty. Accordingly, we find that a cold posterior, tempered by
a power greater than one, often more honestly reflects our beliefs about
aleatoric uncertainty than no tempering -- providing an explicit link between
data augmentation and cold posteriors. We show that we can match or exceed the
performance of posterior tempering by using a Dirichlet observation model,
where we explicitly control the level of aleatoric uncertainty, without any
need for tempering.
