Paper: Smoothing for exponential family dynamical systems (ICLR-2024)

Reviewer comment (Experiments):
The second concern is the limited results. Having derived long technical details, with the approximated ELBO, the Lagrangian-based optimization and the additional algorithm, the manuscript should at least provide results proportional to the technical development. In the reviewer’s opinion, the evaluation of the model is somehow short (learning of two synthetic systems (pendulum and chaotic scenario) plus analysis on convergence). It would have been preferable to see experiments proving the feasibility of the model with long time data or similar, especially because it is indicated that the proposed ELBO delivers a computational cost equivalent to mean-field approximations without such crude factorizations.

Author response:
Dear reviewer,

Thank you for taking the time to provide feedback on the manuscript.  Motivated by concerns of the limited results, we have included an additional **real-data experiment** in order to demonstrate the efficacy of the ideas presented applied to learning a dynamical system from neural data.  Motivated by other reviewer suggestions, we have also removed the vBF section in favor of using the additional space to add more details and have a more cohesive story.  Additionally, we respond to your questions below

1) In this case, the ELBO would not be tight in the general case; but we find it reassuring that the plugin predictive approximation is exact for the linear Gaussian state-space model, and we have tried to include that in the text to aid in building intuition. We have evaluated the tightness empirically in Fig.1. And considering the Kalman smoother and particle filter marginal likelihoods for the example linear and nonlinear dynamical systems, we can see our ELBOs (purple dashed lines) are close to the marginals. 

2) This is a good point. The objective ELBO defined by Eq.14 is still a lower bound of some marginal likelihood by its form. However, it corresponds to a prior different from (approximating) the original one defined by the generative model. Many other approaches in fact have to use approximation too, and thus implicitly introduced a new prior/marginal as well.

3) Theoretically it leads to an ELBO of a different marginal as we mentioned in 2) and in turn might increase or decrease the variational gap. Empirically, we have not found this to be a problem in the experiments performed.

Thank you again for your feedback.  We feel that the additional experiment and approximation details motivated by your questions have helped increase the quality of our manuscript greatly.  We hope that you could consider adjusting your score if you find it warranted given the extra improvements.

Thanks for your time,

The authors

Reviewer comment (Experiments):
The motivation for the method is unclear. For example, the paper states early on that forwards and backwards factorizations have different analysis properties, but this is not mentioned again. Similarly, it is claimed that a certain factorization underuses dynamics and is inefficient, but this is not demonstrated experimentally. There are no results showing that the method overcomes hurdles faced by previous methods.

Author response:
Thank you for your question.  We could have made the point clearer; we were trying to convey that the forward factorization and its ELBO makes it natural to design an inference network that amortizes $q(z_t | z_{t-1})$; however, amortizing $q(z_t | z_{t+1})$ and then evaluating L_B would be comparatively more difficult since we would have to use a Monte Carlo approximation of  $E[p(z_t|z_{t-1})]$, and additionally the KL between those distributions would also require a further Monte Carlo approximation. We have tried to explain this further in the revised manuscript.  However, what we do evaluate is $\hat{\mathcal{L}}_B$, which after substitution of the plug-in predictive, does not require making a MC approximation of the KL term.  We have tried to expand this point more in the main text.

Reviewer comment (Presentation):
The paper would benefit greatly from reducing content. For example, the vBF connection could be removed to allow for a more thorough discussion and comparison of the dBFS and aBFS algorithms, as well as more comparison with existing work.

Author response:
Dear reviewer,

We would like to thank the reviewer for their extensive feedback and careful reading of our manuscript.  Below, we respond in order of your questions, but before doing so, we would like to point to the **additional real-data experiment** motivated by your feedback about the experimental validation.  Additionally, we have also removed the vBF material as per your recommendation in order to make better use of the available space to unpack certain concepts e.g. the predictive plug-in through an illustrating example.

Reviewer comment (Writing):
Throughout the paper, parameters (e.g., $\nu, \lambda$) in equations or objectives do not always appear on the right-hand side (e.g., $\lambda$ and $\theta$ in (17-18)). It would be better to consistently indicate which terms are indexed by which parameters.

Author response:
> Why does L have different parameters in (10) vs (11)?

This is an inconsistency on our side, thank you. We have fixed this in the revised manuscript. We tried to denote ELBOs where a parametric form for the approximation was not assumed yet using $\mathcal{L}(q)$.

Reviewer comment (Experiments):
From the presented set of experiments, it is hard to understand how dBFS compares with aBFS, how useful aBFS really is relative to the baselines, and how aBFS and vBF would scale to more complex problems, indicating room for improvement and unrealized potential in the paper's current form.

Author response:
> 3) how aBFS and vBF would scale to more complex problems. I think there is a lot of room for improvement here, and thus think the paper has a lot of unrealized potential in its current form.

Thank you for this question, it has helped motivate us to apply aBFS to real data in order to validate its efficacy.  We hope the additional experiment applying aBFS to analyzing a more complex real dataset will help position the paper better.

Reviewer comment (Theory):
Is it possible to elaborate on what it means to be a "plug-in" predictive distribution? Are we just saying that we will take the thing that is supposed to be \( q^- \) and plug in \( E_{q(z_{t-1})}[p_\theta(z_t|z_{t-1})] \) instead? Is this different than a posterior predictive distribution?

Author response:
> Is it possible to elaborate on what it means to be a “plug-in” predictive distribution? Is this different than a posterior predictive distribution?

Yes, exactly. It was intended as an acting posterior predictive distribution for a single-step into the future. We name it as we do “plug-in” $\bar{q}$ as an approximation to $E_{q_t}[p(z_t | z_{t-1})]$ (the best exponential family approximation as measured by the forward KL).

Reviewer comment (Reproducibility):
Is it possible to state the number of total parameters that each of the models in the experiments had? Having significantly fewer parameters relative to something like L-VAE could be interesting to demonstrate the model’s effectiveness.

Author response:
> is it possible to state the number of total parameters that each of the models in the experiments had?

Certainly, we have added these in the experiments section.