We would like to thank the reviewer for engaging in the discussion.
First of all, we must clarify that we share the same the notion of epistemic uncertainty; it is indeed given by the gap between $p$ and $\hat p$, and can be alleviated with more training samples. In that sense, for a fixed $p$, the epistemic uncertainty is only a function of the model $\hat p$. We apologize if stating epistemic uncertainty is a property of the model created confusion, but such characterization is also common in the literature (see e.g. the discussion in the introduction in [3]).
While the general definition of epistemic uncertainty might be simple on the surface, actually quantifying this gap does involve assumptions about the model. The same reference quoted by the reviewer [1], states in section 2.1
> To capture EU [epistemic uncertainty], the learner must be able to express uncertainty about $\theta$ [model parameters], which can be accomplished by a (second-order) probability distribution over (first-order) distributions $\theta$.
Conformal prediction methods pose no assumptions on the underlying machine learning model and typically leverages a single predictor/scoring function. In that setting, there is no clear way to represent the uncertainty over the model parameters (or more generally over the model's hypothesis space) and capture epistemic uncertainty, as also highlighted in [2]:
> Machine learning methods for probability estimation, i.e., for training a probabilistic predictor, often commit to a single hypothesis learned on the data, thereby ignoring epistemic uncertainty.
As we mentioned in our rebuttal, that is why we focused on the data aleatoric uncertainty, since quantifying epistemic uncertainty would require assumptions about the model hypothesis space that, at least for now, are not directly connected to any conformal prediction method, and thus are outside the scope of this work.
We share the reviewer's appreciation for the goal of connecting conformal prediction to other notions of uncertainty, and we would argue that relating it to the data aleatoric uncertainty, as we have in this paper, is a valid contribution and a solid first step towards that goal. Further, we would like to emphasize that there are important contributions to uncertainty quantification without epistemic and aleatoric decomposition, and conformal prediction is a notable example. We see our work within this line of research, and as we showed in the paper, our results already proved useful in practical applications, namely conformal training and incorporation of side information.
We hope to have addressed the reviewer's concerns around this topic.
### Extra references
[3] Bengs, Viktor, Eyke Hüllermeier, and Willem Waegeman. "Pitfalls of epistemic uncertainty quantification through loss minimisation." Advances in Neural Information Processing Systems 35 (2022): 29205-29216.