How Tempering Fixes Data Augmentation in Bayesian Neural Networks

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.