Do-calculus

Do-calculus is a set of mathematical rules devised by Judea Pearl in 1995 to determine whether causal effects can be identified from observational data under specific assumptions encoded in a causal graph. It provides a systematic method for transforming expressions involving the do-operator (representing interventions) into expressions involving only observable probabilities, enabling the identification of causal relationships.

Do-calculus is a set of mathematical rules devised by Judea Pearl in 1995 to determine whether causal effects can be identified from observational data under specific assumptions encoded in a causal graph. It provides a systematic method for transforming expressions involving the do-operator (representing interventions) into expressions involving only observable probabilities, enabling the identification of causal relationships. P(y \mid \mathrm{do}(x), z, w) = P(y \mid \mathrm{do}(x), w) \quad \text{if } Y \perp\!\!\!\perp Z \mid X, W \text{ in } \mathcal{G}_{\overline{X}} This rule allows the removal of irrelevant observations ( Z ) if they are d-separated from Y given X and W in the graph where incoming edges to X are removed. P(y \mid \mathrm{do}(x), \mathrm{do}(z), w) = P(y \mid \mathrm{do}(x), z, w) \quad \text{if } Y \perp\!\!\!\perp Z \mid X, W \text{ in } \mathcal{G}_{\overline{X}\, \underline{Z}} This rule permits replacing an intervention ( \mathrm{do}(z) ) with an observation ( z ) if Y and Z are d-separated in the graph where outgoing edges from Z and incoming edges to X are removed. P(y \mid \mathrm{do}(x), \mathrm{do}(z), w) = P(y \mid \mathrm{do}(x), w) \quad \text{if } Y \perp\!\!\!\perp Z \mid X, W \text{ in } \mathcal{G}_{\overline{X}\, \overline{Z(W)}} This rule removes irrelevant interventions ( \mathrm{do}(z) ) if Y and Z are d-separated in a graph modified to block paths through Z .

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