Potential Issue with get_sepset Function Failing to Identify Valid Separating Sets for Conditional Independence

Start with the get_sepset entry point and compare its result with the CIT and FCI runs described in the issue. Reproduce the AU06, AU12, AU14, AU15, and Reward example at significance level 0.05, then verify that the returned sepset agrees with the brute-force result [AU06, AU14, AU15].

Dear developers, thanks for this great library.

Package version:

0.1.3.8

Description

I am a newbie in the field. But I encountered a problem that led me to suspect there might be an issue with the get_sepset function. Specifically, it seems that the function may fail to identify valid sepsets that make two nodes conditionally independent in certain scenarios.

From my understanding, the current implementation appears to rely exclusively on path exploration to determine sepsets. This approach can lead to incorrect results when there are no direct or indirect paths between the nodes but conditional independence still exists for some sets.

Problem Details

Firstly, I tested the conditional independence where the separation set is empty using the CIT class. The resulting p-value between AU12 and Reward was 2.3553502981887853e-05, which means AU12 and Reward should be connected with an edge. This was indeed the case when I applied the FCI algorithm with a feature set including only AU12 and Reward. Then, I ran the FCI algorithm again with the features AU06, AU12, AU14, AU15, and Reward. The resulting graph, shown below, is as expected; conditioning on the other features might remove edges, as we know from the FCI algorithm's mechanism.

However, when I used the get_sepset method, it returned an empty set. Whereas the correct sepset should be [AU06, AU14, AU15] (or the nodes that correspond to them).

Note: I figured out the actual sepset using CIT class and brute forcing the sepsets. The results are shown below.
Note2: I run the FCI algorithm with significance level 0.05

Node1, Node2, sepset:

Reward AU12_r []
kci 2.3553502981887853e-05
Reward AU12_r [' AU14_r']
kci 0.00015905749617084464
Reward AU12_r [' AU15_r']
kci 1.685509670779073e-05
Reward AU12_r [' AU06_r']
kci 8.263130655117301e-05
Reward AU12_r [' AU14_r', ' AU15_r']
kci 0.0014881913935198554
Reward AU12_r [' AU14_r', ' AU06_r']
kci 0.0014911663188149626
Reward AU12_r [' AU15_r', ' AU06_r']
kci 0.003424638498105481
Reward AU12_r [' AU14_r', ' AU15_r', ' AU06_r']
kci 0.06554509205235681