Do operator

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The do operator is used in causal inference to denote an intervention. Given random variables X,Y, we write \Pr(Y=y \mid \mathit{do}(X=x)) to mean the probability that Y=y given we intervene and set X to be x. In some texts, this is abbreviated to \Pr(y\mid\hat x) (this notation assumes that the random variables corresponding to the individual values are clear from context). The notation \Pr_x(y) is also used.

In general \Pr(Y=y \mid \mathit{do}(X=x)) is not the same as conditioning on X=x, i.e. \Pr(Y=y \mid X=x). Note also that in the expression \mathit{do}(X=x), the subexpression X=x does not mean the event where the random variable X takes on the value x, i.e. the event \{\omega\in\Omega : X(\omega) = x\}. Thus, inside a do operator, the standard notational convention of probability theory does not hold. To stress the point, suppose the event X=x can be specified in another way, such as by the event Z=z. In this case, since X=x and Z=z are exactly the same set, the probabilities involving them, such as \Pr(X=x) vs \Pr(Z=z) and \Pr(Y=y \mid X=x) vs \Pr(Y=y \mid Z=z), should all be the same, but I don't think \Pr(Y=y \mid \mathit{do}(X=x)) and \Pr(Y=y \mid \mathit{do}(Z=z)) need be the same (check this).

The do operator is used extensively in the do calculus.

History

Pearl: "An equivalent notation, using \mathit{set}(x) instead of \mathit{do}(x), was used in Pearl (1995a). The \mathit{do}(x) notation was first used in Goldszmidt and Pearl (1992) and is gaining in popular support. Lauritzen (2001) used P ( y \mid X \leftarrow x). The expression P(y \mid \mathit{do}(x)) is equivalent in intent to P(Y_x = y) in the potential-outcome model introduced by Neyman (1923) and Rubin (1974) and to the expression P[(X = x) \mathbin{\Box\!\!\rightarrow} (Y = y)] in the counter-factual theory of Lewis (1973b)."[1]

References

  1. Judea Pearl. Causality. p. 70, footnote 2