Thank you for your response and active engagement in the rebuttal discussion. We are delighted that you are positive and interested in our work. In the following, we would like to clarify your comments:
### Observations
> In addition, is it, e.g., $\mathbf{t}^{1} = (1,5,7,9)$, $\mathbf{t}^{2} = (2,5,6)$, where $\mathbf{t}^{1}$ and $\mathbf{t}^{2}$ are two data samples, and the lengths may be different?
Indeed, instances or data samples of a TPP can have different lengths and could, for example, look like $(1,5,7,9)$ and $(2,5,6)$. In other words, both the arrival times and number of points are random. This defining property of a TPP is introduced in the background section 2.1 lines [59-61]. Further, we would like to highlight the notational difference in your response. With the superscript $\mathbf{t}^{(n)}$, *we* refer to an event sequence at the $n$-th diffusion step, where $\mathbf{t}^{(n)} \sim \lambda_n$ and not to different samples from the data distribution.
> Clarify ``temporal embedding followed by a sequence embedding'' and the input to the network at each stage (can you maybe add code).
The temporal and sequence embedding were explained in Sec. 3.3 lines [178-183] with the additional sentence in our last response and further depicted in Fig. 3. Here, we will additionally present some background and slightly informal explanation of the embedding types for TPPs:
**Temporal (Positional) embedding:** Applying temporal embeddings, e.g., log-transform and trigonometric functions (similar to the positional embeddings of Transformer [3]), to event times is a common practice for TPPs (see Sec. 3.1 of [1]). With a temporal embedding, we refer to a function $\mathbb{R}_{>0} \to \mathbb{R}^d$, that maps a time, e.g., $t_i \in \mathbf{t}$ independently to a $d$-dimensional embedding space. Then for each ${t}_i$, we have a feature vector $\mathbf{c}_i$ encoding its temporal information.
**Sequence embedding:** To attain an event embedding (also sometimes called context embedding) $\mathbf{e}_i$ for each event $i$ that also incorporates information about all other events, we apply a three-layered CNN with dilation and circular padding on the sequence of temporally embedded event times $(\mathbf{c}_1, \cdots, \mathbf{c}_K)$. Ultimately, a global sequence embedding $\bar{\mathbf{e}}$ is attained by computing the average over $(\mathbf{e}_1, \cdots, \mathbf{e}_K)$.
The code will be published upon acceptance, but we present a simplified pseudo-code of our implementation of the sequence embedding here:
```
# temporal embedding
time_embedding = self.time_encoder([x_n.time, x_n.tau]) # Kxd
# event embedding
event_embedding = self.cnn(time_embedding) # Kxd
# sequence embedding
seq_embedding = event_embedding.mean(dim=0) # d
```
As described in Sec. 3.3 line [185; 196] and Fig. 3, ```time_embedding``` $\mathbf{c}_i$ and ```event_embedding``` $\mathbf{e}_i$ are the inputs to the classifier and ```seq_embedding``` $\bar{\mathbf{e}}$ is the input to the MLPs parameterizing the intensity function.
### Set distinction
> Is it also possible to have a figure (Distinction between sets A,...,F)?
We might be wrong but believe that Fig. 1 (right) and 2 (left) are what you are looking for. Fig. 2 presents the case distinction of the sets **A**,...,**F** with a Venn diagram as the logical relations between $\mathbf{t}^{(0)}$, $\mathbf{t}^{(n-1)}$ and $\mathbf{t}^{(n)}$. Further, in Fig. 1, the denoising process illustrates the case distinction for one specific example of $\mathbf{t}^{(0)}$ and $\mathbf{t}^{(n)}$. Please note that the illustrated posterior intensity can be separated into the sets **B-E**: The base intensity relates to $\lambda^{(E)}(t)$, while the highest peaks (Diracs) refer to $\lambda^{(B)}(t)$, i.e., the points that are in $\mathbf{t}^{(0)}$ and $\mathbf{t}^{(n)}$. The medium peaks and lower peaks (thinned Diracs) represent $\lambda^{(D)}(t)$ and $\lambda^{(C)}(t)$, respectively. Please let us know if this addresses your point, and we will, for the camera-ready version, add an additional reference to the denoising process in Fig. 1 in the discussion of the reverse process in Sec. 3.2.
### Approximation
> Approximating Dirac deltas with Gaussians may not work as well...
A Gaussian is the standard approximation of the Dirac delta function and can, in the limit $\sigma \to 0$, perfectly approximate it [2, 4]; a property that is not true for many approximations of other TPP models (e.g., parametric distribution for the intensity function of *RNN*, *Trans* and *TriTPP*). Furthermore, the KL divergence between Dirac and our Gaussian is directly minimized to ensure the best possible approximation.
### References
1. Citation 38 in the paper
2. Ghatak, Ajoy, et al. "The dirac delta function." Quantum Mechanics: Theory and Applications (2004)
3. Citation 42 in the paper
4. Saichev et al. "Chapter 1: Basic definitions and operations", Distributions in the Physical and Engineering Sciences (1997)