reviews: 
Comments: The paper proposes a new way to approximate the posterior distribution over parameters for a parametric model $p(x;\theta)$. 

The approach:
* defines 2 multinomial distributions:
   *  $y(\\{\theta_k\\})$ over the index $k$ of $K$ parameter samples $\theta_k$ as a function of the model $p(x,\theta)$ and noise distribution; 
   * $\hat{y}(\\{\theta_k\\})$ as a function of the parameterised approximate posterior $q_\phi(\theta)$ and the noise distribution;
* $\hat{y}$ is trained to match $y$ under cross entropy loss;
* at optimality $q_\phi(\theta)$ matches the true posterior $p(\theta|x)$ (subject to the choice of the parametric family $q_\phi$).

Results compare the proposed method to typical amortised variational inference (maximising the ELBO) and a more similar method, showing improved outcomes on those datasets.
Strengths: * The proposed method makes sense and appears straight-forward to implement. 
* The method appears general to problems with a parametric model where $p(x,\theta)$ can be computed.
* Comparison to SNIS is revealing and clear.
Weaknesses: Overall I like the paper, find it to be of high quality and soundness and the proposed method is a clever trick that converts a difficult problem into a simpler classification-type task. Areas in which the paper could improve (from good to excellent) are: 

**Clarity**: as someone familiar with this field but not immersed in it, the paper could be clearer to read. As specific examples:
* [033, 078] From the outset, the paper refers to *latent variables*, which often refer to variables that are one-to-one with data samples, whereas the term *parameters* typically relates to global variables of the model, i.e. common across all samples. (Of course parameters are often treated as unobserved, hence latent, variables). I appreciate that this may be subjective, but I find it to be fairly universal and a useful distinction, so this phrase put the wrong starting point in my mind. I was wondering throughout when "parameters of the model" get updated and only when I reached the experiments did it become clear that it is about parameters and it made more sense. I would suggest referring to "parameters" (treated as latent variables).
* The preamble to Eq 2 is not directly relevant and could be more clear: labels y are constructed analytically and the approximate posterior is trained by learning to predict them. The backstory of predicting k* doesn't seem so relevant (as it is in NCE, say) so this bit threw me off until I realised that.
* [202-213] This part stands out as less precise relative to preceding arguments. The method trains one ratio of ratios to match another ratio of ratios using cross entropy loss. Reference to "classification being too easy" and "punishing" seems out of place and unclear. The issue seems to be the relative range of values in the softmax function (+/- numerical instability?), which is mentioned but blended with loose wording. This part could be improved, particularly as computing these ratios/choosing the negative sampling distribution is the crux and potential pitfall of ratio estimation.
* [161] Ensuring $\pi$ covers the true posterior - I can't see how this is "guaranteed", unless pi is diffuse, which sounds problematic for ratios.
* Experiments: to compute a posterior, you need a prior, but the priors don't seem clearly specified.
* Fig 1: how are some log probability values positive?
* GARCH experiment: The descriptions of $\alpha_1$ and $\beta_1$ are rather vague.

**Analysis/Experiments**: For such a widely applicable method, the experiments to showcase its efficacy/analyse performance etc seem thin. 
* Understanding how well it works in higher dimensions is of interest, where potentially finding a negative distribution close to the true posterior could be less straightforward.
* Understanding how the methods compare as number of data points increases is of interest to know when the method is most applicable.
* Understanding how the methods compare when the posterior is mis-specified is of interest to understand robustness (mode seeking may be preferable to distribution covering?).

Minor
* [134] it would aid clarity to explicitly say optimisation is "with respect to $\phi$"
* [134] it would be much clearer to state labels y "(from Eq 2)".
* [370] "a different observation" of what?
* [421-424] the paper does well to be relatively self-contained, this part could be made more clear to the less familiar reader.
Recommendation: 10
Reviewer confidence: 4
Soundness/correctness: 4
Clarity: 2