Dealing with Hidden Confounders

If we assume that all genuine causes are measured, the causal validation step of TCDF consisting of attention interpretation and CQII validation should in theory only discover correct causal relationships (according to the data). Namely, cases 2b, 3b and 4b from Section 4.2.1 then all arose because of a measured confounder. A time series Xi that was correlated with time series Xj because of a confounder would not be labeled as true cause by CQII, since only the presence of the confounder would be needed to predictXj.

However, our CQII approach might discover incorrect causal relationships if there exist hidden con- founders, i.e. confounders that are not included in the dataset. This section describes how TCDF can successfully discover the presence of a hidden confounder with equal delays to its e↵ects Xi and Xj (case 4b from Section 4.2.1). But, we also discuss that TCDF will probablynot detect the presence of a hidden confounder when the hidden confounder has unequal delays to its e↵ects (case 2b and 3b).

As is shown in Table 1, most temporal causal discovery methods cannot deal with hidden confounders in a dataset. Method ANLTSM can only deal with hidden confounders that are linear and instantaneous. The authors of TiMINo claim to handle hidden confounders by staying undecided instead of inferring any (possibly incorrect) causal relationship. Lastly, tsFCI handles hidden confounders by including a special edge type (Xi$Xj) that shows thatXi is not a cause ofXj and thatXj is not a cause ofXi. From this, the authors conclude that there should be a hidden confounder that causes bothXi andXj.

TCDF can discover this Xi $Xj relation in specific cases by applying CQII. Based on cases 2, 3 and 4, we can distinguish three reasons why two time series are correlated: existence of a causal relationship, presence of a measured confounder or presence of a hidden confounder. (We exclude the possibility of a spurious correlation). If there is a measured confounder, CQII should discover that the confounder’s e↵ects

Xi and Xj are just correlated and not causally related. If there is a 2-cycle, CQII should discover thatXi causesXjwith a certain delay and thatXj causesXiwith a certain delay. If there is ahiddenconfounder of

Xi andXj, CQII will find thatXi is a true cause ofXj and vice versa. Namely, if CQII uses an intervened distribution for Xi, the loss _L of network_Nj to predict Xj will probably increase, since both the hidden confounder and the intervenedXi cannot give away any information about the future values ofXj.

When the delay from the confounder toXi is smaller than the delay toXj (case 3b), TCDF will, based on the temporal precedence assumption, discover an incorrect causal relationship from Xi to Xj. More specifically, TCDF will discover that the delay of this causal relationship will be equal to the delay from the confounder toXi minus the delay from the confounder toXj. Figure 9a shows an example of this situation. The same reasoning applies when the delay from the confounder toXi isgreater than the delay toXj (case 2b), since Xi and Xj are just interchanged such that TCDF will discover an incorrect causal relationship fromXj toXi.

3

1 4

X

X

X

(a) Example showing that TCDF will incorrectly discover a causal relationship fromX2 toX3 when the delay from hidden confounderX1to e↵ectX2is

smaller than the delay fromX1 toX3.

0 0

4 4

X

X

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(b) Example showing that TCDF will discover a

2-cycle betweenX2 andX3 where both delays equal

0, such that it concludes that there should exist a hidden confounder betweenX2 andX3.

Figure 9: Example showing how TCDF deals with hidden confounders (denoted as a square). A black square indicates that the hidden confounder is discovered, whereas a grey square indicates that the hidden confounder is not discovered by TCDF. Black edges indicate causal relationships that will be included in the learnt temporal causal graph_GL. Grey edges indicate that these causal relationships are not included in_GL.

However, TCDF willnot discover a causal relationship when the hidden confounder has equal delays to its e↵ectsXiandXj (case 4b), and can even conclude that there should be a hidden confounder that causes bothXiandXj. Because the confounder has equal delays toXiandXj, both the delay from the discovered causal relationship fromXitoXjwill be 0, and the delay from the discovered causal relationship fromXj to

Xiwill be 0. The zero delays give away the presence of a hidden confounder, since there cannot exist a 2-cycle where both time series have an instantaneous e↵ect on each other. Recall that an instantaneous e↵ect means that there is an e↵ect within 1 measured time step. If both time series cause each other instantaneously, there will be an infinite causal influence between the time series within 1 time step, which is impossible. Therefore, TCDF will in this case conclude thatXiandXj are not causally related, and that there exists a hidden confounder betweenXiand Xj. Figure 9b shows an example of this situation.

The advantage of our approach is that TCDF not only concludes that two variables are not causally related, but can also detect the presence of a hidden confounder. This might be useful information for further research.

ˆ X1 2Xˆ22Xˆ32Xˆ24 ... Xˆ216 Output Hidden Hidden Hidden Input X1 1 X21X13 X14 X110 X116 0.8 0.3 1.2 0.4 1.2 0.4 0.2 0.2 1.6 0.2 0.2 1.6 1.6 1.6 1.3 1.3 1.3 1.3 1.3 1.3 1.3 1.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 f= 23 f= 22 f= 21 f= 20

Figure 10: An example that discovers the delay between cause X1 and target X2, both having T = 16. Starting from the top convolutional layer, the algorithm traverses through the path with the highest kernel weights. Eventually, the algorithm ends in input valueX10

1 , indicating a delay of 16 10 = 6 time steps.