Chain

As explained in Causal Inference for Data Science

A chain is a three-node directed path in which the arrows point consecutively in the same direction, as in X→Y→Z or X←Y←Z.

A chain is a three-node directed path whose arrows point consecutively in one direction: X→Y→Z or X←Y←Z. In a causal graph, the middle node transmits association between the endpoints. section 7.1.4

Rain, wet streets, and dirty floors

Let X mean rain, Y a wet street, and Z shoes making the floor dirty. Because Z depends on Y, knowing Y makes the details of X irrelevant: P(Z|Y,X) = P(Z|Y). Thus X and Z are conditionally independent given Y, and conditioning on Y blocks the path. section 7.1.4

A chain can carry correlation between its endpoints even when conditioning on the middle node removes it. In synthetic data, X and Z had correlation 0.32 overall, but correlations near zero after separating observations by Y. section 7.1.4

Use the chain pattern when reasoning about paths in a DAG:

• Conditioning on the intermediate node blocks the chain between the endpoints. section 7.1.5
• Distinguish chains from forks and colliders: forks have arrows pointing away from a common cause, while colliders have arrows pointing into a common effect. section 7.1.4
• When seeking d-separation, block every relevant path with a chain or fork node, but avoid conditioning on a collider or its descendant unless another variable closes the opened path. section 7.1.6

Table 7.2 Chain DAG

DAG	Equations

Inspect the chain DAG and its structural equations for the rain–street–floor example. section 7.1.4

Listing 7.1 (R) Creating a synthetic dataset with the structure of a chain

set.seed(1234)
n <- 10000
x <- rbinom(n, 1, 0.5)
y <- x * rbinom(n, 1, 0.3) + (1-x)*rbinom(n, 1, 0.7)
z <- y * rbinom(n, 1, 0.1) + (1-y)*rbinom(n, 1, 0.9)
cor(x, z)

Run the R data-generating process that creates a synthetic chain. section 7.1.4

Listing 7.2 (Python) Creating a synthetic dataset with the structure of a chain

from numpy.random import seed, binomial
from numpy import corrcoef
seed(1234)
n = 10000
x = binomial(1, 0.5, n)
y = x * binomial(1, 0.3, n) + (1 - x) * binomial(1, 0.7, n)
z = y * binomial(1, 0.1, n) + (1 - y) * binomial(1, 0.9, n)
corrcoef(x, z)[0][1]

Run the Python equivalent of the synthetic chain simulation. section 7.1.4

How Causal Inference for Data Science covers it

How deep it goes

Defines the pattern, derives conditional independence, and tests it with synthetic correlations, while deferring broader graph factorization details.

Examples

Demonstrates rain, wet streets, dirty floors, Bernoulli synthetic data, and R and Python conditional-correlation checks.

What it assumes

Assumes familiarity with variables, correlation, probability, and DAG notation while developing path-conditioning intuition from simple examples.

What you can do after

Evaluate causal graphs by recognizing chains and selecting intermediate variables that block their paths.

How it connects

Connects chains to DAGs, causal paths, conditional independence, d-separation, forks, colliders, and descendants.

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Related concepts

• pathkind of · A path is an ordered sequence of connected elements representing a route through a graph, structure, or space.
• conditional independenceproduces · Two variables are conditionally independent given specified information when their conditional joint distribution factors into the product of their conditional distributions.
• pipelineused by · A pipeline is an ordered or coordinated sequence of connected processing stages or tasks that passes data, work, or results toward an outcome.

This explainer was generated by Manning's knowledge graph from the text of Causal Inference for Data Science. The section links point to the parts of the book it draws on.