Editing Causality (section)

=== Causal calculus ===
When experimental interventions are infeasible or illegal, the derivation of a cause-and-effect relationship from observational studies must rest on some qualitative theoretical assumptions, for example, that symptoms do not cause diseases, usually expressed in the form of missing arrows in [[causal graphs]] such as [[Bayesian network]]s or [[path analysis (statistics)|path diagrams]]. The theory underlying these derivations relies on the distinction between ''conditional probabilities'', as in <math>P(cancer|smoking)</math>, and ''interventional probabilities'', as in <math>P(cancer|do(smoking))</math>. The former reads: "the probability of finding cancer in a person known to smoke, having started, unforced by the experimenter, to do so at an unspecified time in the past", while the latter reads: "the probability of finding cancer in a person forced by the experimenter to smoke at a specified time in the past". The former is a statistical notion that can be estimated by observation with negligible intervention by the experimenter, while the latter is a causal notion which is estimated in an experiment with an important controlled randomized intervention. It is specifically characteristic of [[quantum mechanics|quantal phenomena]] that observations defined by incompatible variables always involve important intervention by the experimenter, as described quantitatively by the [[Observer effect (physics)|observer effect]].{{Vague|date=June 2016}} In classical [[thermodynamics]], [[thermodynamic process|processes]] are initiated by interventions called [[thermodynamic operation]]s. In other branches of science, for example [[astronomy]], the experimenter can often observe with negligible intervention.

The theory of "causal calculus"<ref name=Pearl>Pearl, Judea (2000). ''[https://books.google.com/books?id=wnGU_TsW3BQC Causality: Models, Reasoning, and Inference] {{Webarchive|url=https://web.archive.org/web/20210831203927/https://www.google.com/books/edition/_/wnGU_TsW3BQC |date=31 August 2021 }}'', Cambridge University Press.</ref> (also known as do-calculus, [[Judea Pearl]]'s Causal Calculus, Calculus of
Actions) permits one to infer interventional probabilities from conditional probabilities in causal [[Bayesian network]]s with unmeasured variables. One very practical result of this theory is the characterization of [[confounding variables]], namely, a sufficient set of variables that, if adjusted for, would yield the correct causal effect between variables of interest. It can be shown that a sufficient set for estimating the causal effect of <math>X</math> on <math>Y</math> is any set of non-descendants of <math>X</math> that <math>d</math>-separate <math>X</math> from <math>Y</math> after removing all arrows emanating from <math>X</math>. This criterion, called "backdoor", provides a mathematical definition of "confounding" and helps researchers identify accessible sets of variables worthy of measurement.