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We challenge the longstanding assumption that the mean-field approximationfor variational inference in Bayesian neural networks is severely restrictive,and show this is not the case in deep networks. We prove several resultsindicating that deep mean-field variational weight posteriors can inducesimilar distributions in function-space to those induced by shallower networkswith complex weight posteriors. We validate our theoretical contributionsempirically, both through examination of the weight posterior using HamiltonianMonte Carlo in small models and by comparing diagonal- to structured-covariancein large settings. Since complex variational posteriors are often expensive andcumbersome to implement, our results suggest that using mean-field variationalinference in a deeper model is both a practical and theoretically justifiedalternative to structured approximations.

Liberty or Depth: Deep Bayesian Neural Nets Do Not Need Complex Weight Posterior Approximations