X = f_{X} (U_{X}), A = f_{A} (U_{A}), M = f_{M} (A, X, U_{M}), Y = f_{Y} (M, X, U_{Y}) .

The bidirected edge

A \leftrightarrow Y

permits dependence between

U_{A}

and

U_{Y}

; apart from this pair the exogenous variables are mutually independent. The basic potential outcomes are

M (a, x) = f_{M} (a, x, U_{M}), Y (m, x) = f_{Y} (m, x, U_{Y}),

and the natural nested outcome is

Y (a) = Y (M (a, X), X)

The exogenous independences imply that the whole family

{M (a, x) : a, x}

is independent of

{X, A, Y (m, x) : m, x}

, while

X

is independent of

A

and the basic response potentials. The bidirected edge means that

A

and

Y (m, x)

need not be independent. Useful observed-counterfactual consequences include

M (a) ⊥ ⊥ A ∣ X, M ⊥ ⊥ Y (m) ∣ A, X .

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