  Bayesian neural networks (BNNs) are a principled approach to modeling
predictive uncertainties in deep learning, which are important in
safety-critical applications. Since exact Bayesian inference over the weights
in a BNN is intractable, various approximate inference methods exist, among
which sampling methods such as Hamiltonian Monte Carlo (HMC) are often
considered the gold standard. While HMC provides high-quality samples, it lacks
interpretable summary statistics because its sample mean and variance is
meaningless in neural networks due to permutation symmetry. In this paper, we
first show that the role of permutations can be meaningfully quantified by a
number of transpositions metric. We then show that the recently proposed
rebasin method allows us to summarize HMC samples into a compact representation
that provides a meaningful explicit uncertainty estimate for each weight in a
neural network, thus unifying sampling methods with variational inference. We
show that this compact representation allows us to compare trained BNNs
directly in weight space across sampling methods and variational inference, and
to efficiently prune neural networks trained without explicit Bayesian
frameworks by exploiting uncertainty estimates from HMC.
