  We investigate the benefit of treating all the parameters in a Bayesian
neural network stochastically and find compelling theoretical and empirical
evidence that this standard construction may be unnecessary. To this end, we
prove that expressive predictive distributions require only small amounts of
stochasticity. In particular, partially stochastic networks with only $n$
stochastic biases are universal probabilistic predictors for $n$-dimensional
predictive problems. In empirical investigations, we find no systematic benefit
of full stochasticity across four different inference modalities and eight
datasets; partially stochastic networks can match and sometimes even outperform
fully stochastic networks, despite their reduced memory costs.
