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We establish the first mathematically rigorous link between Bayesian,variational Bayesian, and ensemble methods. A key step towards this it toreformulate the non-convex optimisation problem typically encountered in deeplearning as a convex optimisation in the space of probability measures. On atechnical level, our contribution amounts to studying generalised variationalinference through the lense of Wasserstein gradient flows. The result is aunified theory of various seemingly disconnected approaches that are commonlyused for uncertainty quantification in deep learning -- including deepensembles and (variational) Bayesian methods. This offers a fresh perspectiveon the reasons behind the success of deep ensembles over procedures based onparameterised variational inference, and allows the derivation of newensembling schemes with convergence guarantees. We showcase this by proposing afamily of interacting deep ensembles with direct parallels to the interactionsof particle systems in thermodynamics, and use our theory to prove theconvergence of these algorithms to a well-defined global minimiser on the spaceof probability measures.

A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods