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do-calculus

nounAlso: do-operator, do(x), causal calculus, back-door criterion, front-door criterion

In plain wordsA set of mathematical rules, devised by Judea Pearl, for working out the effect of deliberately changing something using only observational data plus a diagram of what causes what.

  1. A set of three rules, due to Judea Pearl, that use a causal graph (structural causal modelstructural causal model, n.: A precise description of what causes what in a system — a diagram of arrows plus equations — that predicts what would happen if you intervened, not just what tends to occur together.) to rewrite interventional quantities — P(y | do(x)), the distribution of Y when X is set to x by intervention, as distinct from the observational P(y | x) — step by step until, where possible, only ordinary observational probabilities remain. The rules say when observations may be added or deleted and when actions may be exchanged with observations or deleted, based on conditional-independence patterns in modified versions of the graph. If repeated application removes every do-operator, the causal effect is identifiable from observational data; the back-door and front-door adjustment formulas are special cases. The calculus is complete: when it cannot identify an effect, no method can without further assumptions. It underlies algorithmic identification in causal-inference software and gives a formal language for distinguishing seeing from doing.

    Using do-calculus, we showed the recommendation's effect on purchases is identifiable through the front-door path.

P(y∣do(x))=∑zP(y∣x,z) P(z)(back-door adjustment for Z)P(y \mid do(x)) = \sum_{z} P(y \mid x, z)\,P(z) \qquad \text{(back-door adjustment for } Z\text{)}

Origin

Introduced by Judea Pearl in 'A Probabilistic Calculus of Actions' (Conference on Uncertainty in Artificial Intelligence, 1994) and 'Causal Diagrams for Empirical Research' (Biometrika, 1995), building on his 1993 back-door criterion. Its completeness was established in 2006 by Yimin Huang and Marco Valtorta and, independently, by Ilya Shpitser and Pearl. Pearl received the 2011 Turing Award for work including this calculus for causal reasoning.

Related

See also
  • causal inferencecausal inference, n.: Working out whether one thing actually causes another, rather than merely occurring alongside it — for example, whether a new feature made users stay longer or just happened at the same time.
  • structural causal modelstructural causal model, n.: A precise description of what causes what in a system — a diagram of arrows plus equations — that predicts what would happen if you intervened, not just what tends to occur together.
  • confounding variableconfounding variable, n.: A hidden third factor that influences two things at once and makes them look linked. Ice-cream sales and drownings both rise in hot weather; neither causes the other.
  • counterfactualcounterfactual, n.: A statement about what would have happened if something had been different — 'had I left earlier, I would have caught the train'. AI uses counterfactuals to explain decisions and to measure causes.
  • Bayesian networkBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence.
  • conditional probabilityconditional probability, n.: The chance of something once you already know that something else is true — for example, the chance of rain given that the sky is cloudy.

Sources

  1. Pearl, J. (1994). A Probabilistic Calculus of Actions. Proceedings of the 10th Conference on Uncertainty in Artificial Intelligence (UAI 1994). arXiv:1302.6835. arxiv.org/pdf/1302.6835
  2. Pearl, J. (1995). Causal Diagrams for Empirical Research. Biometrika, 82(4).

Mentioned in

Bayesian networkBayesian network, n.: A diagram of causes and effects with probabilities attached — say, how rain and a sprinkler can each make grass wet — that lets a computer update its beliefs about hidden causes when it sees evidence., causal inferencecausal inference, n.: Working out whether one thing actually causes another, rather than merely occurring alongside it — for example, whether a new feature made users stay longer or just happened at the same time., confounding variableconfounding variable, n.: A hidden third factor that influences two things at once and makes them look linked. Ice-cream sales and drownings both rise in hot weather; neither causes the other., counterfactualcounterfactual, n.: A statement about what would have happened if something had been different — 'had I left earlier, I would have caught the train'. AI uses counterfactuals to explain decisions and to measure causes., Judea PearlJudea Pearl, prop. n.: An Israeli-American computer scientist who gave AI the mathematics for reasoning under uncertainty and, later, for telling cause and effect apart from mere correlation., Simpson's paradoxSimpson's paradox, n.: A statistics trap in which combining groups reverses what each group shows: a treatment can do better than another for men and for women separately, yet worse when everyone is counted together., structural causal modelstructural causal model, n.: A precise description of what causes what in a system — a diagram of arrows plus equations — that predicts what would happen if you intervened, not just what tends to occur together.