Official comment by the authors
Thanks for your reply!
We would like to respond to your comments and clarify them.
> For instance, equation (2) is In my opinion, stated without any commentary beyond it corresponding to the probability of sampling $\mathbf{n}$ and $P_0$ being the normalization constant.
We agree that we keep the description of equation (2) short. However, due to the page limit, we decided to keep the preliminaries relatively compact and refer to the appendix for more details.
> Unless I'm mistaken, I do not see the need to denote by n both the number of different colors in the urn as well as the number of marbles per color and then again the sample size?
Thanks for your question. We want to clarify that we denote the number of different colors, the number of marbles per color, and the sample size by $n$ because in our version of the urn model (partition model) they all have the same value.
What often leads to confusion is the number of subsets. Note that, at most, there can be as many subsets as there are elements in the set we want to partition, namely if each element is placed in a separate subset. Since each color stands for one subset, there are a maximum of $n$ colors. Our paper follows previous work [a], which introduced $K$ as the number of subsets where $K \leq n$.
Further, in our preliminaries section of the MVHG distribution and equation (2), we initially follow the notation of [4] and state that there are $m_k$ marbles of color $k$ in the urn. In the case of a partition, each subset can contain, at most, as many elements as there are elements in the set we want to partition. Thus, we set the number of marbles for each color to $n$, i.e., $\forall k\in[K]: m_k=n$.
Hence, we decided to keep $n$ as a central part of our notation instead of introducing additional notation that denotes the same value.
[a] Mansour, T., & Schork, M. (2015). Commutation relations, normal ordering, and Stirling numbers. CRC Press.
> As a paper on differentiable subset sampling I feel like perhaps the means by which you render such an operation differentiable should come earlier, with more discussion, instead of simply being relegated to the appendix.
We decided to focus the main part of our paper on the idea of the two-stage procedure, which we view as our main contribution, and keep the technical details brief. In our opinion, this improves the clarity of our paper and helps in understanding the core part of our contribution. However, we see your point, and without space constraints, we would have added more discussion on the differentiability aspect of the proposed formulation.