Originality: The framework is in general novel. A similar method exists for discrete utilities but not continuous ones. The paper contributes to the loss of calibrated VI for continuous utility. Quality: The work has comparability high-quality. Clarity: In general, the wring is clear. Each step of the work is well-motivated explained. a) However, it would be great to have an explicitly related work section. Due to lack of related work section, related work is either briefly cited such as [14] or not event mentioned, I believe that more related work in approximate inference needs to be mentioned (such as much recent work from Tom Rainforth). b) The linearization in 3.2 should be described to make the paper self-contained. c) The estimation of M_1 using VI should be expaned as well. d) Small things, such as Figure 1 should be put earlier and the caption should be more informative if it is used in the introduction. e) The tasks in the experiments also need more description. Also the recommended one also need to motivate why a continuous utility is needed. For this task, I believe that discrete unity can be used as well. Significance: The technical contribution is incremental but I believe the whole framework can be very useful and significant for many applications.

This paper proposes to adapt the variational inference process to better capture the posterior regions relevant to decision-making with continuous utilities (when enumeration is no longer tractable). The result is an importance-weighted ELBO. Using nested Monte Carlo integration and double reparametrization the authors are able to provide the tools to convert continuous unbounded losses into utilities that guarantee optimal calibration. By formulating the problem as a joint optimization procedure of decisions and approximation parameters, the authors showcase how this procedure can be straightforwardly carried out by automatic variational inference. This adds to the significance of his work. Overall, while this paper might lack in originality, it is clear and well written. The proposed tools are sound and were thoroughly studied in terms of calibration. Question: why can we assume smoothness / differentiability of the expected utility ?

Originality: The paper builds on ideas developed by Lacoste-Julien et al. (2011) that were introduced to bridge Bayesian decision theory with approximate inference in a meaningful and useful way. The paper takes these ideas and makes them applicable in continuously-valued settings so long as the losses are bounded. For inference, it uses a variation of 'black box' type variational inference schemes. Quality: The paper makes an interesting contribution. However, it is undesirable that the losses must be bounded. Is there a fundamental reason why one cannot extend the proposed methodology to unbounded losses? One potential way of extending/adapting the method that comes to mind is using recent advances in loss-based/PAC/generalized Bayesian posteriors, see e.g. Bissiri, Holmes & Walker, '16 or the work on Generalized VI. The idea there would be to side-step the conversion into utilities altogether. Unless I misunderstood something fundamental, I believe this could be possible by designing a kind of compound loss: adding the negative log likelihood (which is in fact a type of loss itself) to a decision-making-driven loss would generate a new compound/additive loss for the parameter of interest. This in turn would produce an exact & coherent (Gibbs/generalized) posterior in the sense of Bissiri, Holmes & Walker, '16. The variational approximation of such a generalized posterior could then be seen as a form of Generalized VI. If there are fundamental reasons why strategies of this form would be inappropriate/would not be achieving the same goal, it would be good to know what makes them conceptually fundamentally different from utility-based approaches. Clarity: Overall, the paper is well-written and clearly understandable. For me, the one exception to this is the explanation for why one should calibrate the utilities such that their infimum is 0 (lines 107-116). Even after multiple readings, I could not understand where the conclusion that 'for optimal calibration we should use utilities such that inf u(y,h) = 0' comes from. Significance: The paper clearly makes a significant contribution by extending utility/loss calibration into continuously-valued settings. The fact that the loss needs to be bounded however is problematic and should ideally be addressed. EDIT: I was very happy with the reviewer response as it managed to answer my biggest questions convincingly. Accordingly, I am happy to raise my score to a 6 provided that the explanation of the 0-infimum rationale is expanded upon in the main paper in the same way it was expanded upon in the rebuttal -- the extra page will provide the space necessary. Reading your paper once again, I also noticed that using the non-linear loss to utility transformation the authors propose in eq. (3) actually already DIRECTLY corresponds to the kind of compound-loss I was alluding to in my original response. I failed to notice this in my first reading but would encourage the authors to include a short elaboration on this connection in a future version of the paper.