Venue: ICLR 2021 Conference

Reviews:

final_score: rating: 6: Marginally above acceptance threshold
review_content: This paper proposes an interesting point process named diffusion modulated cox processes, which generalizes the stochastic intensity to a stochastic differential equation. The variational inference method looks sound. 

Pros:
+ The generalization of SDE-type intensity is novel. The proposed stochastic variational inference makes sense. Especially the neural network solution is meaningful and will have an impact on the learning of point processes. 
+ Good empirical performance and analysis. 

Cons:
- I strongly recommend the authors to further improve the presentation of the current draft. Sometimes the notations are not consistent, like $\mathbb{E}$ and $E$, $\mathbb{R}$ and $R$. The definition of some of the notations are not very clear, such as Eq(3), what is the definition of the conditional expectation? Is it the expectation of the intensity function given all the information about the process?  Besides, please give a brief introduction of the mathematical background, such as the smoothing posterior. It would also be better if the authors can give some examples to illustrate the advantages of the proposed diffusion modulated cox process.
- Please discuss the influence of the non-negativity of the intensity function. The paper claims that $h$ is non-negative and thus can be an identity function, which is false.
review_title: The idea of the SDE-modulated cox is interesting, but the writing needs to be improved.

final_score: rating: 7: Good paper, accept
review_content: The paper provides a stochastic variational inference method for
approximating posterior path measures for doubly-stochastic Poisson processes
conditioned on realisations of paths of the Poisson process. The intensity
process is modelled as the solution to a diffusion stochastic differential
equation. The authors compare their method experimentally to the numerical
solution of the associated system of stochastic partial differential
equations using the finite element method.

The paper, although technically difficult, is well written and clearly
presented. While I cannot validate the mathematical content in detail, the
paper seems technically correct. It deals with a well-defined problem in a
difficult mathematical setting. 

The idea of using variational inference for approximating the smoothing
posterior density seems well-founded. The method is experimentally validated
on simulated and real data.

In summary, though my knowledge of point process theory is not sufficient to
evaluate all aspects of the paper in detail, I find the paper to be a solid
contribution worthy of publication at ICLR.
review_title: Solid contribution on point processes and variational inference

final_score: rating: 7: Good paper, accept
review_content: The paper under review proposes a variational inference procedure for a specific class of Cox processes whose intensity is derived from a stochastic differential equation. The methodology relies on a restriction of candidate solutions the the subset for which the drift depends on $x_t$, $N_t$ and $t$; the drift is then modelled with a neural network. By simulating from the candidate model, a sample average approximation of the ELBO is used to compute a stochastic gradient descent algorithm, optimize the bound and thus estimate non-parametrically the drift.

The paper is clear and well written with original and significant contribution. The paper would benefit from clarifications about the following points:


## Major points
* It is not clear how equation (13) is a subset of the solution of (2) in general more parameters implies more degree of freedom and thus the drift in (13) could look more general than $b(x)$ (the use the $\bar{b}$ notation suggesting it refers more to (11) than (2)). This probably comes from the fact that all variables are indexed by $t$ but more details would be nice here.
* The proposed framework requires to specify both $b$ and $\sigma$ a priori. It would be nice to explain what is the impact and restrictions of doing so. For instance, the first experiment set $\sigma$ to $1$ and the second to $1.1$ without further justification. Could the author(s) elaborate(s) on this point? Similarly, the authors use a very flexible model (NN) for $\bar{b}$, so the main restriction comes from the choice of the `target' $b$: the impact of this choice should be discussed in more detail.
* Similarly to the previous points, in the methodology, the link function $h$ must be set a priori (and its uncertainty is not taken into account). Would it be possible to have an inference methodology estimating not only the drift but also $h$ (and $\sigma$?) at the same time? If not, is it possible to explain why?


## Minor points
* p.1 l.1: the reference for Cox processes is not the 'original' work.
* p.2: 'in particular, we show that...' either 'that' or 'how'.
* p.2: VBSP has never been defined at this stage of the paper.
* p.3: Wiener.
* p.3: the notation $\mathbb{E}^\dagger$ is undefined.
* p.3: defined the indicator function that you are using.
* p.3: shouldn't the $x$ in the definition of $\bar{v}(x_t)$ be bold?
* p.3: which 'section'? Simply 'above'?
* p.3: $f$ denotes first bounded measureable functions and then twice-differentiable ... Use two different notation to avoid confusion.
* In general, use eqref for equations.
* p.5: We call $Q^\star$ as the VB ... Defined the meaning of VB.
* p.5: specify the class of parametric functions used in Sutter et al.
* p.6: the notation $\psi_t'$ is not defined. The link with equation (13) could be made clearer.
review_title: The paper is clear and well written with original and significant contribution.

Meta-review: The paper presents a stochastic variational inference method for posterior estimation in a Cox process with intensity given by the solution to a diffusion stochastic differential equation. The reviewers highlight the novelty of the approach. Some of the concerns with regards to clarity have been addressed by the authors satisfactorily. 

However, an important issue of the approach is that of estimating model parameters, which the authors do not address explicitly by simply referring to that as the task of the modeller. I believe this is an important issue and, although some of the parameters can be estimated along with the neural network parameters, this has not been shown empirically. Along a similar vein, the paper only presents results on a single real dataset (the bike-sharing dataset), which questions the applicability of the approach and no other baseline method is presented. At the very least, the authors should have provided an objective evaluation to other doubly stochastic point process models, e.g. based on Gaussian processes, where modern stochastic variational inference algorithms have been presented. 

Decision: Reject