The do operator is used in causal inference to denote an intervention. Given random variables , we write to mean the probability that given we intervene and set to be . In some texts, this is abbreviated to (this notation assumes that the random variables corresponding to the individual values are clear from context). The notation is also used.
In general is not the same as conditioning on , i.e. . Note also that in the expression , the subexpression does not mean the event where the random variable takes on the value , i.e. the event . Thus, inside a do operator, the standard notational convention of probability theory does not hold. To stress the point, suppose the event can be specified in another way, such as by the event . In this case, since and are exactly the same set, the probabilities involving them, such as vs and vs , should all be the same, but I don't think and need be the same (check this).
The do operator is used extensively in the do calculus.
Pearl: "An equivalent notation, using instead of , was used in Pearl (1995a). The notation was first used in Goldszmidt and Pearl (1992) and is gaining in popular support. Lauritzen (2001) used . The expression is equivalent in intent to in the potential-outcome model introduced by Neyman (1923) and Rubin (1974) and to the expression in the counter-factual theory of Lewis (1973b)."
- Judea Pearl. Causality. p. 70, footnote 2