NeurIPS 2019
Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center
Paper ID: 7231 Kalman Filter, Sensor Fusion, and Constrained Regression: Equivalences and Insights

### Reviewer 1

The paper considers an elegant new perspective on Kalman Filters, by demonstrating how it might be considered a sensor fusion problem by absorbing the process error as a block diagonal term of the sensor error covariance matrix. Further, the paper builds on this to formulate time-series data prediction from sensors as a regression based on empirical covariance estimates of the sensor error. This tool is then used to formulate prediction of flu activity from proxy measurements. The formulation is well motivated and the results are good. The supplement offers further explanation of how covariance shrinkage might be viewed as being within this framework. This particular version yields good results in the use case demonstrated. Further discussions on how the new formulation can lead to sensor selection (based on significance) are useful, but preliminary. A minor nitpick is that the authors speculate that non-linear models could have offered further benefits, but do not complete the evaluation.

### Reviewer 2

Rebuttal acknowledged, thank you for the additional clarifications. --- Originality: I believe that the findings of Section 2 are well-known given a Bayesian / Gaussian viewpoint of the KF (c.f. [1] and [2]). Indeed, given a flat prior for $x_{t+1}$ (i.e., Gaussian with "infinite" variance), we have two independent observations: - the influence of the past (prediction term) - the influence of the current measurement (filtering term) both have Gaussian likelihood. So the posterior density of $x_{t+1}$ is proportional to a product of three Gaussian-shaped terms. The two different ways in which these terms can be folded into each other (using standard Gaussian conjugacy rules) lead to Thm 1. I believe that the linear-algebraic formulation the authors use just hides the fact that we are multiplying Gaussian PDFs in different ways. On the other hand, I think that the reformulation of Section 3 is less straightforward and perhaps of larger interest. Just like the connection between the KF and linear-Gaussian models opened up many new possibilities, I believe the authors' reformulation may lead to practical improvements, such the ones outlined in Section 5. Quality: Generally speaking, the paper is sound and rigorous. The experimental evaluation of Section 4 is well explained. The results are somewhat underwhelming: it is not clear that the proposed method does significantly better than competing ones. I believe that the paper would benefit from having some of the new ideas outlined in Section 5 explored in the flu-nowcasting applications. E.g., maybe adding L1-regularization helps? Clarity: Generally speaking, the paper is well-written and easy to follow. The supplementary material is also clear and well-structured. Table 1 is not easy to parse. I would suggest presenting the results in a bar plot instead. - line 67: "no a process" -> typo - line 87: "intuitive extend"' -> typo [1]: Faragher, Understanding the basis of the Kalman filter via a simple and intuitive derivation, 2012 [2] Särkkä, S. Bayesian Filtering and Smoothing, 2013

### Reviewer 3

I am afraid that I struggle to see what is new in this paper and what its significance is. As stated by the authors themselves the main result in Theorem 1 “is elementary”, like an exercise for a linear systems course. There has been quite a lot of work on viewing the Kalman filtering problem as a convex optimisation problem, see e.g. https://web.stanford.edu/~boyd/papers/pdf/rt_cvx_sig_proc.pdf I have a feeling that this view will help the authors a fair bit in developing their ideas further and find relationships with more existing work. A detail: I would recommend the authors to reconsider the extremely narrow definition you are using for the word sensor fusion. Sensor fusion is a much wider term, see e.g. its Wikipedia entry https://en.wikipedia.org/wiki/Sensor_fusion Academically a lot of research related to sensor fusion is published at the yearly fusion conference https://fusion2019.org/ From just a very quick look at these sites it should be clear that sensor fusion is much broader that the particular linear equation (8) in the current manuscript.