reviews: 
Comments: The paper considers a variational inference approach that builds upon classifying ‘noisy’ samples from a learned proposal distribution as negative samples, similar to a noise contrastive estimation approach, and where ‘ground truth soft labels’ depend on a self-normalised importance sampling estimate for the log ratio of the target density relative to the proposal density. In contrast to a classical ELBO minimizing the reverse KL to the target, the suggested approach appears to better approximate the forward KL to the target. Numerical experiments on benchmark Bayesian inference problems illustrate that suggested approach yields te better coverage probabilities and yields to approximations with less bias for the posterior mean estimate.
Strengths: The paper addresses the classic and important question of Bayesian inference, particularly for the challenging setting where the approximate posterior should be close to the target with respect to the forward KL to yield well calibrated inferences. The approach avoids previous objectives that yield mass-covering behaviour, but are more challenging to optimise, particular in high dimensions. The suggested objective is now as far as I am aware. It is also very interesting that the suggested approach can be used to construct a previously constructed objective (SNIS-fKL), but with a gradient estimator that has zero variance in the idealised setting when the approximate posterior coincides with the true posterior. Experiments do indeed show that the suggested method improves the coverage and leads to approximate posterior that are closer to the target in terms of the forward KL.
Weaknesses: The empirical verification seems a bit limited. In particular, the only baseline (beyond a standard ELBO) is the SNIS-fKL objective. However, as the paper shows, they have the same expected gradients. It is thus not clear how well the suggested approach compares to other (approximate) Bayesian methods that have different averaged gradients, e.g. [1-3]. 

Also as the suggested approach has a higher computation complexity requiring sampling and density-evaluations of K particles, evaluations with a simple IWAE approach instead of the ELBO baseline appears appropriate. 

The authors consider (in Section 2.1) that they have access to samples from the true posterior. However, is having access to samples from the true posterior not the aim of the method? In (4), is $y$ equal to $\hat{y}$ as in Algorithm 1 when one does not have true posterior samples? 

It is not clear why eq. (4) is the right objective to optimise when one does not have access to the true labels $y$. Does optimizing it correspond to some proper scoring rule or divergence [4] that are minimized for this objective? 

Can you clarify why the optimal classifier recovers the true posterior and that the true posterior is equal to the approximate variational distribution as per eq. (6)? The ‘optimal classifier parameterized as in eq. (3)’ depends (apparently?) on the number of samples $K$ from the proposal and on the proposal distribution. Do all the arguments on page 3 from line 141 not depend on these choices? I understand that the classifier is optimal with respect to a cross-entropy loss if it recovers the Bayesian posterior [4]. But is this optimality even realisable in the suggested form (3)?

The experiments target posteriors that are not that high-dimensional (<=200). Does the method scale to high-dimensional problems? Does it overcome scalability issues commonly faced when optimizing the forward KL or for importance sampling?


[1] Dieng, Adji Bousso, et al. "Variational Inference via $\chi $ Upper Bound Minimization." Advances in Neural Information Processing Systems 30 (2017).\
[2] Li, Chengrui, et al. "Forward $\chi^ 2$ Divergence Based Variational Importance Sampling." arXiv preprint arXiv:2311.02516 (2023).\
[3] Naesseth, Christian, Fredrik Lindsten, and David Blei. "Markovian score climbing: Variational inference with KL (p|| q)." Advances in Neural Information Processing Systems 33 (2020): 15499-15510.\
[4] Mohamed, Shakir, and Balaji Lakshminarayanan. "Learning in implicit generative models." arXiv preprint arXiv:1610.03483(2016).
Recommendation: 5
Reviewer confidence: 4
Soundness/correctness: 2
Clarity: 3