Testable implication

As explained in Causal AI

A testable implication is an observable statistical consequence that follows from a model's assumptions and can be checked against data.

A testable implication is an observable statistical consequence of a causal model’s assumptions. A proposed DAG can imply constraints on the joint distribution, which data can test rather than merely use for prediction. section 4.5

Testable implications arise in two closely related forms:

• Conditional independencies follow from d-separation. They can be tested when the required variables are observed. section 4.5 section 4.5.2
• Functional constraints can remain testable when some variables are latent, extending model checks beyond fully observed DAGs. section 4.5 section 4.5.2
• Unobserved variables reduce the available checks: a constraint involving an unavailable variable cannot be tested directly. section 4.5

Smoking, cancer, and unobserved genetics

A DAG for smoking and cancer includes tar buildup, lung cancer, smoking, and genetics. If genetics is unobserved, tests involving it are unavailable, but the model still implies the observable conditional independence (C ⊥ T | S). Functional constraints provide another route for testing the DAG despite the latent variable. section 4.5

Figure 4.20 A causal DAG representing smoking’s effect on cancer. The variable for genetics (D) is gray because it is unobserved in the data, so we can’t run tests for conditional independencies involving D. However, we can test other types of constraints.

Use the DAG figure to distinguish observed variables from the gray latent genetics node and identify which model constraints remain testable. section 4.5

How Causal AI covers it

How deep it goes

Explains conditional independence and functional constraints with graphical intuition, but stops short of detailed functional-constraint mathematics.

Examples

Uses a smoking–cancer DAG with unobserved genetics, tar buildup, and an observable conditional independence.

What it assumes

Assumes causal DAGs and conditional independence, while motivating d-separation and latent-variable testing through graphical intuition.

What you can do after

Diagnose causal models by enumerating observable independence tests and considering functional constraints for latent-variable settings.

How it connects

Connects causal DAGs with latent variables, d-separation, conditional independence, local Markov properties, and functional constraints.

liveBookmanning.com

Read Causal AI on liveBook Get the book at manning.com

Related concepts

• causal graphical modelproduced by · A causal graphical model is a probabilistic or structural model whose graph represents causal relationships and whose mechanisms support reasoning about interventions.
• structural causal modelproduced by · A structural causal model represents causal relationships among variables through equations that determine each variable from other variables and exogenous factors.

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