10.2 The Calculus of Causation

117

a great stumbling block in the development of quantitative causal thinking that no

mathematical notation existed to capture the results of an intervention. ⁴ This was

provided by Judea Pearl’s do-calculus. ⁵

Thus,y vertical bar d o left parenthesis x right parenthesis comma z right parenthesisy|do(x), z) signifies the probability ofupper Y equals yY = y given thatupper XX is held constant

atxx andupper Z equals zZ = z is observed. Unlike the associational models of traditional statistical

analysis, causal models (sometimes called structural models) can be used to predict

how the probabilities of events would change as a result of external interventions,

whereas associational models assume that conditions remain the same. Pearl (2001)

has given an analogy to visual perception: a precise description of the shape of a

three-dimensional object is useful and sufficient for predicting how that object will

be viewed from any angle, but insufficient for predicting how the shape might change

if it is squeezed by external forces, which requires information about the material

from which the object is made and its Young’s, bulk, etc. moduli.

Pearl has given three rules of causal inference, ⁶ which allow sentences concerning

interventions to be transformed into others concerning observations only. The causal

model is directed as a cyclic graph upper GG and upper X comma upper Y comma upper Z and upper WX, Y, Z and W are disjoint subsets of

variables. The rules are:

Rule 1

(insertion/deletion of observations)

upper W 2 left parenthesis y 1 t 1 comma y 2 t 2 right parenthesis equals upper W 1 left parenthesis y 1 t 1 right parenthesis upper W 1 left parenthesis y 2 t 2 right parenthesis commaP(y|(x), z, w) = P(y|(x), w) if (Y ⊥Z|X, W)G ¯X .

(10.1)

Rule 2

(action/observation exchange)

upper W 2 left parenthesis y 1 t 1 comma y 2 t 2 right parenthesis equals upper W 1 left parenthesis y 1 t 1 right parenthesis upper W 1 left parenthesis y 2 t 2 right parenthesis commaP(y|(x), do(z), w) = P(y|(x), z, w) if (Y ⊥Z|X, W)G ¯X Z .

(10.2)

Rule 3

(insertion/deletion of actions)

upper W 2 left parenthesis y 1 t 1 comma y 2 t 2 right parenthesis equals upper W 1 left parenthesis y 1 t 1 right parenthesis upper W 1 left parenthesis y 2 t 2 right parenthesis commaP(y|(x), (z), w) = P(y|(x), w) if (Y ⊥Z|X, W)G ¯X,Z(W) .

(10.3)

In words, Rule 1 states that if a variable upper WW irrelevant to upper YY is observed, then the

probability distribution of upper YY will not change provided variable set upper ZZ blocks all the

paths fromupper WW toupper YY after having deleted all paths leading toupper XX; Rule 2 states that if a

setupper ZZ of variables blocks all paths fromupper XX toupper YY, thenMathID27(x) is equivalent to observingxx

(conditional onupper ZZ); and Rule 3 states thatMathID30(x) can be removed fromMathID31P(y|(x) whenever

there is no causal path fromupper XX toupper YY, i.e.,MathID34P(y|(x) = P(y). Huang and Valtorta (2006)

have shown that these three rules are complete, in the sense that if a causal effect is

identifiable, a sequence of operations exists that transforms the causal effect formula

into one that only includes observational quantities.

4 For an account of how statistics was able to approach relative causal effects, see Reiter (2000).

5 Pearl (1994).

6 See also Pearl (2019).