University of Michigan - Ann Arbor

Daisy, a dog adopted by Dana and his wife, had been having a hard time accepting training/supervision. After several weeks Daisy was still having occasional “accidents” inside the house. 
Dana and his wife by chance observed that bringing foster kittens in the house can stop Daisy from producing "accidents" in the house. According to Dana, the foster kittens remained with them for three weeks, and Daisy the dog did not break her training at all during that period. 
Why would this happen? Is it because of the foster kittens that Dana and his wife accidentally brought into the house?


Senario Setup: Daisy and the kittens

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Causal diagram for Daisy and the kittens

The first P term is the outcome of the Daisy experiment. It represents the probability of successful house training ($$Y = 1$$), given that we set X to 0 (do not introduce other pets) but set the mediator $$M$$ to $$M_1$$ (the value it would have if we had introduced them). The counterfactual $$M_1$$ needs to be computed accordingly: different dogs might have different needs for supervision.
The second P term is the probability of successful house training of the dog under the condition with no other pets. Here we set $$X = 0$$ to reflect that no other pets are present, and $$M = M_0$$ to reflect that we do not introduce the mediator.

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