  While Bayesian neural networks (BNNs) provide a sound and principled
alternative to standard neural networks, an artificial sharpening of the
posterior usually needs to be applied to reach comparable performance. This is
in stark contrast to theory, dictating that given an adequate prior and a
well-specified model, the untempered Bayesian posterior should achieve optimal
performance. Despite the community's extensive efforts, the observed gains in
performance still remain disputed with several plausible causes pointing at its
origin. While data augmentation has been empirically recognized as one of the
main drivers of this effect, a theoretical account of its role, on the other
hand, is largely missing. In this work we identify two interlaced factors
concurrently influencing the strength of the cold posterior effect, namely the
correlated nature of augmentations and the degree of invariance of the employed
model to such transformations. By theoretically analyzing simplified settings,
we prove that tempering implicitly reduces the misspecification arising from
modeling augmentations as i.i.d. data. The temperature mimics the role of the
effective sample size, reflecting the gain in information provided by the
augmentations. We corroborate our theoretical findings with extensive empirical
evaluations, scaling to realistic BNNs. By relying on the framework of group
convolutions, we experiment with models of varying inherent degree of
invariance, confirming its hypothesized relationship with the optimal
temperature.
