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.

Definition and purpose

Causal queries involving interventions (e.g., $P(y \mid \mathrm(x))$) 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.

The three rules of ''Do''-calculus

The rules apply to a causal graph $\mathcal$ and assume the Markov condition holds:

Rule 1: Insertion/deletion of observations

:$P(y \mid \mathrm(x), z, w) = P(y \mid \mathrm(x), w) \quad \text Y \perp\!\!\!\perp Z \mid X, W \text \mathcal_$

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.

Rule 2: Action/observation exchange

:$P(y \mid \mathrm(x), \mathrm(z), w) = P(y \mid \mathrm(x), z, w) \quad \text Y \perp\!\!\!\perp Z \mid X, W \text \mathcal_$

This rule permits replacing an intervention ($\mathrm(z)$) with an observation ($z$) if $Y$ and $Z$ are *d*-separated in the graph where outgoing edges from $Z$ are removed.

Rule 3: Insertion/deletion of interventions

:$P(y \mid \mathrm(x), \mathrm(z), w) = P(y \mid \mathrm(x), w) \quad \text Y \perp\!\!\!\perp Z \mid X, W \text \mathcal_$

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

Applications

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

Completeness

The ''do''-calculus is considered complete: if repeated application of the rules cannot eliminate the ''do''-operator, the causal effect is not identifiable. This result was formalized in 2006 by Huang, Valtorta, Shpitser, and Pearl.

Criticism

Critics have pointed out that other frameworks, such as structural equation modeling (SEM) or Bayesian networks, may offer more intuitive approaches to causal inference for certain applications. These methods often emphasize parameter estimation rather than identifiability, which can be more relevant for applied research.

References

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Statistics
Inference
Causal inference