Getting the correct identifiability expression for the example from Shpitser and Pearl (2008)

[neildhir]

Hi there!

First, excellent package, it is a joy to use.

I was wondering if you could give some input on your example that you provide in the notebook here: https://github.com/y0-causal-inference/y0/blob/main/notebooks/Identification.ipynb - specifically the one regarding figure 3(a) from Complete Identification Methods for the Causal Hierarchy by Shpitser and Pearl (2008) page 1951.

In the notebook (see screenshot below), the final line of that example returns the intervention in terms of conditionals from $P$.

This is not the same expression as the one provided by the authors, which is (as you yourselves put it): $$P_x(y_1,y_2) = \sum_{w_2} P(y_1,w_2) \sum_{w_1} P(y_1 \mid x , w_1)P(w_1)$$.
I am trying to understand why y0 is not returning the same expression as in the original example, and if you could steer me in the right direction to accomplish that. I.e. can additional computations be invoked to simplify the expression into the same form as derived by Shpitser and Pearl?

MWE:

from y0.graph import NxMixedGraph
from y0.dsl import W1,W2,X,Y1,Y2


GG3a = NxMixedGraph.from_edges(
    directed=[
        (W1, X),
        (X, Y1),
        (W2, Y2),
    ],
    undirected=[
        (W1, Y1),
        (W1, W2),
        (W1, Y2),
        (W2, X),
    ],
)

query = Identification.from_expression(graph=GG3a, query=P(Y1 @ X, Y2 @ X))
identify(query)

That output is different from the paper and your own notebook. Hence I am trying to understand what is happening here (or more likely where I am going wrong).

Thanks