In this paper, we present a general framework for distilling expectations with respect to the Bayesian posterior distribution of a deep neural network, significantly extending prior work on a method known as "Bayesian Dark Knowledge." Our generalized framework applies to the case of classification models and takes as input the architecture of a "teacher" network, a general posterior expectation of interest, and the architecture of a "student" network. The distillation method performs an online compression of the selected posterior expectation using iteratively generated Monte Carlo samples from the parameter posterior of the teacher model. We further consider the problem of optimizing the student model architecture with respect to an accuracy-speed-storage trade-off. We present experimental results investigating multiple data sets, distillation targets, teacher model architectures, and approaches to searching for student model architectures. We establish the key result that distilling into a student model with an architecture that matches the teacher, as is done in Bayesian Dark Knowledge, can lead to sub-optimal performance. Lastly, we show that student architecture search methods can identify student models with significantly improved performance.
Contributions:
The paper considers the distillation of a Bayesian neural network as presented in [Balan et al. 2015]
The main contribution of the paper is the extension of [Balan et al. 2015] to apply to general posterior expectations instead of being restricted to predictions. 
A second contribution of the paper is the finding that restricting the architecture of the student network to coincide with the teacher can lead to suboptimal performance and this can be mitigated by expanding the student's architecture using architecture search.
Originality/Significance:
I want to discuss the result regarding the generalization of the posterior expectation (section 3.1). To my knowledge this is novel, however, I am failing to see the importance of this result. The paper mentions two cases as motivations: to calculate the entropy and the variance of the marginal. The  problem with these two examples is that it is unclear why these are important and they are not used in the experiments anywhere. Their use should be motivated and the performance of the distillation should be properly evaluated in the experiments.
The result regarding architecture search is interesting, but it should be expanded and explained more to be a main contribution.
Clarity:
The paper generally understandable, and well written, but it could be better organized. It should expand on the motivation and the architecture search, since these are key components of the paper. The figures are not legible. Even fully zoomed in, they are difficult to read.
Overall assessment:
The paper has some interesting ideas, but it lacks motivation and significant results.
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Response to the rebuttal:
Thank you for the detailed reply.
> A primary motivation for generalising posterior expectations is to help quantify model uncertainty. Indeed, the expectation of the predictive entropy is an important quantity that is distinct from the entropy of the posterior predictive distribution. For instance, the difference between these two quantities is exactly the BALD score used in active learning [1]. ...
BALD would be problematic to use with this framework due to computational costs. The main benefit of distillation is the reduced computational cost at inference time. But training itself is still expensive. In BALD, the bulk of the computational cost is fitting the model after each new observation. Distillation does not provide a speedup here.
If the method indeed works well in an active learning setting, it would be interesting to see experiments showcasing this result.
> - The result regarding architecture search is interesting, but it should be expanded and explained more to be a main contribution.
I was hoping for more discussion/guidance on finding the right architecture or perhaps an algorithm that efficiently optimises the architecture. But I understand that this is more of a future work so I am not holding this against the paper.
I realise that my initial assessment was rather short so I decided to increase my rating and lower my confidence score. I think the paper is borderline, but I am slightly leaning towards rejection due to the insufficient motivation.