For single-hidden-layer ReLU Bayesian neural networks, mean-field Gaussian and Monte Carlo dropout posteriors provably cannot express higher predictive variance between well-separated low-variance regions, and this limitation persists empirically in deep networks despite a universality theorem.
Variational inference: A review for statisticians
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On the Expressiveness of Approximate Inference in Bayesian Neural Networks
For single-hidden-layer ReLU Bayesian neural networks, mean-field Gaussian and Monte Carlo dropout posteriors provably cannot express higher predictive variance between well-separated low-variance regions, and this limitation persists empirically in deep networks despite a universality theorem.