Game 4
"This one introduces a new kind of bias, called 'M-bias' (named for the shape of the graph). [...]
M-bias puts a finger on what is wrong with the traditional approach. It is incorrect to call a variable, like B, a confounder merely because it is associated with both X and Y. To reiterate, X and Y are unconfounded if we do not control for B. B only becomes a confounder when you control for it!" (Pearl, pp. 161โ162)
Game 4 in BayesiaLab
The structure of this example seems simple and can be easily analyzed in BayesiaLab:โ
Given that B is a collider, there is no open path and, thus, there is no effect of X on Y at all.
As a result, nothing needs to be blocked.
However, as Pearl explains, if one were to apply a traditional three-step for a confounder, one might (incorrectly) conclude that B should be controlled for as a confounder.
Let's try this scenario in BayesiaLab and see what happens.
By controlling for B, we inadvertently open up a noncausal path between X and Y, i.e., we are introducing a bias.
The Influence Path Analysis highlights the M-shape, for which this bias is known.
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