

(1) Comparison between LoLiCaP and ICP (Method II from their paper) in three settings: two settings where the assumptions of ICP are met and have either dense or sparse heterogeneity in the environments, and one setting where the assumptions of ICP are violated. [+]

(2) We perform PCMCI [1] on the Lorenz-data set with n=25 [+]


3) We add a LiNGAM comparison with a scaled-student-t noise distribution, once for [+]

4) Moving assumptions of Theorem 2 and 3 to the main paper. [+]

5) Below assumption 1, "It ensures the environment does not act as a confounder between covariates and target.": Can you elaborate how it ensures this?
-> Change the sentence into "it ensures that the noise distribution is the same in all environments, which is a crucial property we test for within our methodology.[+]

6) The definition of causal parents 
 (near equation (1)) is circular:

However, with respect to Assumption 1 your point is certainly valid and we change "There exists a subset 
..." to, "Let 
 be defined as above,..." [+]

 7) below Lemma 1, "the above null hypothesis": which one:

We will specify that we mean  [+]

8) Assumption 2, "where 
 is a polynomial of finite but otherwise arbitrary": I'm not quite sure what this means. Is it that if 
 is a parent, then [+]

 -> Specify

 9) Regarding Assumption 2, it seems that the authors assume that the causal mechanisms of all 
 are the same across environments, but not for 
 variables. Is there a particular reason/motivation for this?

This is well spotted and just a notational mistake. We will add an environment index to those mechanisms.[+]

10) The name "locally linear": -> change to localized?

11) Calling Section 6.1 an "application" is very misleading: We agree. The term application was not intended to refer to a real-world setting and associated data. 

-> We will name the section simply "Network Detection in Dynamical Systems"[+]

12) Re-calibration for alpha: 

-> The experiment leading to Figure 6 in appendix C.1 is not meant as a possible fix to the problem, but is merely supposed to show that both the false positive and negative rate are negatively affected by the wrong calibration. We are happy to make this context clearer.[+]

13)  Include extra example: In addition, we plan to include the following extra example (with additional details): Our task is to identify important factors that drive fluctuations in stock prices and market volatility. For that task we identify important legislation regarding regulations of the stock market. One may assume that both, the distributions of the important factors, as well as the mechanism between factors and stock prices and market volatility can change after an important legislation was put into place. We chose our environments now as the time between those legislations.[+]

14) Finding causal parents of a disease:... -> add mechanism shift relation[+]

15) Finding causes of a mechanism shift (...) samples should be sufficiently far apart in time 

16) Changing distribution of the target noise... can you elaborate what you mean by independence of the environment index  and 
 which seems to depend on 

-> This is indeed confusing, we will remove the index 
 from 
. It was just supposed to indicate that this is the specific 
 we observed together with environment 
, i.e. we observe joint entities [+]

17) Add sketch proof for Thm 1: [+]

18) Add remark about mechanism intervention [+]

19) Check typos they mention [+]

20)"[...] finding a full causal graph is possible in our proposed setting if no covariate is directly affected by the environment index, but only indirectly through changing structural parameters." could be phrased more precisely.

21) The introduction is rather short: Perhaps one could move some of the motivating examples from Section 3 there?

22) Plotting false positive and negative rate in one plot is suboptimal due to the different scaling. It makes Figures 3, 4, 6 rather hard to read.

