Do-calculus

Causal queries involving interventions (e.g., ${\displaystyle P(y\mid \mathrm {do} (x))}$) are considered identifiable if they can be expressed using observational data alone, independent of unmeasured parameters. The do-calculus achieves this by leveraging graphical criteria from directed acyclic graphs (DAGs) to remove do-operators through algebraic manipulations.^[1]

The rules^[2] apply to a causal graph ${\displaystyle {\mathcal {G}}}$ and assume the Markov condition holds:

${\displaystyle P(y\mid \mathrm {do} (x),z,w)=P(y\mid \mathrm {do} (x),w)\quad {\text{if }}Y\perp \!\!\!\perp Z\mid X,W{\text{ in }}{\mathcal {G}}_{\overline {X}}}$

This rule allows the removal of irrelevant observations (${\displaystyle Z}$) if they are d-separated from ${\displaystyle Y}$ given ${\displaystyle X}$ and ${\displaystyle W}$ in the graph where incoming edges to ${\displaystyle X}$ are removed.

${\displaystyle P(y\mid \mathrm {do} (x),\mathrm {do} (z),w)=P(y\mid \mathrm {do} (x),z,w)\quad {\text{if }}Y\perp \!\!\!\perp Z\mid X,W{\text{ in }}{\mathcal {G}}_{{\overline {X}}\,{\underline {Z}}}}$

This rule permits replacing an intervention (${\displaystyle \mathrm {do} (z)}$) with an observation (${\displaystyle z}$) if ${\displaystyle Y}$ and ${\displaystyle Z}$ are d-separated in the graph where outgoing edges from ${\displaystyle Z}$ and incoming edges to ${\displaystyle X}$ are removed.

${\displaystyle P(y\mid \mathrm {do} (x),\mathrm {do} (z),w)=P(y\mid \mathrm {do} (x),w)\quad {\text{if }}Y\perp \!\!\!\perp Z\mid X,W{\text{ in }}{\mathcal {G}}_{{\overline {X}}\,{\overline {Z(W)}}}}$

This rule removes irrelevant interventions (${\displaystyle \mathrm {do} (z)}$) if ${\displaystyle Y}$ and ${\displaystyle Z}$ are d-separated in a graph modified to block paths through ${\displaystyle Z}$.

Do-calculus can be applied to various domains within causal inference such as mediation analysis in decomposing direct and indirect effects.^[3][4] It can be used for meta-synthesis to combine the results from heterogeneous studies.^[3][5]