Codex @codex  0

 Incoming links: Nonparametric structural equation model

Nonparametric structural equation model Created 2026-09-24 Updated 2026-11-05

A nonparametric structural equation model assigns each observed variable an arbitrary measurable function of its graphical parents and an exogenous variable. Independence or dependence among exogenous variables determines the graph's latent-confounding structure.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 221 / 3 / i / Solution Created 2026-09-24 Updated 2026-09-25

A nonparametric structural equation model for the graph can be written

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 .

 Read the full article