Update `intro` folder of Sphinx `examples/` to show intuition and concepts of inducing path

[adam2392]

As a follow-up to #78.

Perhaps we can just use one graph to illustrate the point. Can we use figure 2 from the paper https://arxiv.org/pdf/1104.5617.pdf

Then proceed by:

1. drawing the figure and describing the inducing path and figure
2. By default adjacent nodes have a trivial inducing path (the edge between them)
3. show that inducing paths among non-adjacent nodes relative to L={L1, L2}, S={} is only possible in the figure between X1 and X5.
4. There is no inducing path relative to L and S between (X1, X6) and (X2, X4)
5. By adding X6 to S, then there is now an inducing path relative to L and S among (X2, X4), (X1, X3), (X5, X3)

This walks through the core concepts we've walked through on this PR on a graph that is in a well-known publication and then describes some of the intricacies of the hyperparameters (L and S).

Originally posted by @adam2392 in #78 (comment)

cc: @aryan26roy may be interested :)

[aryan26roy]

3. show that inducing paths among non-adjacent nodes relative to L={L1, L2}, S={} is only possible in the figure between X1 and X5.

@adam2392 there is no inducing path from X1 to X5 with L={L1,L2} and S={}.

[adam2392]

Sorry you're right. There is no primitive inducing path among any nodes initially. This is validated by the final Figure 2c, where the skeleton is exactly the same before/after.

If you set X3 as part of L, then there will be an inducing path between X1 and X5.

[Update] So I guess this could be more complicated, but I think perhaps the story of the example can go:

• look there are no inducing paths except for the trivial ones among adjacent nodes
• now look if we set X3 as a part of L, we can reason that X1 and X5 now has an inducing path of X1, L1, X2, X3, X4, L2, X5.
• now look if we set X6 as part of S, we can reason that other inducing paths are now available since it would open up all colliders that are ancestors of X6.
• conclude with a short snippet of how inducing paths relate to MAGs (we can cover MAGs/PAGs in a separate example at that point)

WDYT?

[aryan26roy]

That sounds good. Just to clarify something, there are a lot of inducing paths other than [X1,X5] which are available with L = {L1,L2,X3}. Adding X6 to S opens up ALL inducing paths, right?

[adam2392]

That sounds good. Just to clarify something, there are a lot of inducing paths other than [X1,X5] which are available with L = {L1,L2,X3}. Adding X6 to S opens up ALL inducing paths, right?

That I don't know. I think it might be overwhelming to give a laundry list to the user. The goal is to essentially show a figure and illustrate what happens when L and S is modified and walk them through 1-2 specific examples of why those paths are now "inducing".

[aryan26roy]

Understood. I think I get what you want.