Functional constraint

As explained in Causal AI

A functional constraint is a testable restriction on an observed probability distribution implied by a causal graph, including one obtained after accounting for latent variables.

A functional constraint is a testable restriction on an observed probability distribution implied by a causal graph. Unlike ordinary d-separation, it can constrain independence between an observed variable and a function of observed variables even when the graph contains latent variables. section 4.5.1

For a graph containing cost C, smoking S, treatment T, lung-cancer status L, and latent D, h(L,C,T) combines P(L | C,S,T) with P(S | C) and sums over S. The graph implies that C and h(L,C,T) should be independent, creating a testable check without observing D. section 4.5.1

A practical testing workflow is:

• Derive constraints algorithmically from a DAG with latent nodes using causaleffect's verma.constraints or Y0's r_get_verma_constraints wrapper. section 4.5.2
• Fit P(L | C,S,T) and P(S | C), then implement h by multiplying corresponding probabilities and summing across S. section 4.5.3
• Attach h values to each data row and inspect independence from C with a box plot or ANOVA F-test. section 4.5.3

Cigarette cost and cancer data

Discretize cigarette cost into low, medium, and high categories, fit naive Bayes models for the two required conditional distributions, calculate h for every C–T–L combination, and compare h across cost levels. Overlapping box-plot distributions support the predicted independence; ANOVA supplies a corresponding p-value. section 4.5.3

Use the causal graph to trace how latent structure produces a functional independence involving observed variables. section 4.5.1

How Causal AI covers it

How deep it goes

Descends from definition to model fitting, summation, row-level construction, visualization, and ANOVA, but defers deeper mathematical derivation.

Examples

Uses cigarette cost, smoking intensity, treatment, lung cancer, fitted probability models, box plots, and ANOVA.

What it assumes

Assumes familiarity with DAGs, d-separation, conditional independence, probability distributions, and basic statistical modeling.

What you can do after

Diagnose causal-model misfit by deriving and statistically checking functional constraints with fitted probability models.

How it connects

Links causal graphs, latent variables, d-separation, conditional independence, causal models, and falsification.

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Related concepts

• observational joint distributionuses · An observational joint distribution is the true joint probability distribution of variables as observed without externally imposed interventions.
• latent variableuses · A latent variable is an unobserved quantity represented or inferred by a model to capture underlying structure in observed data.
• statistical independence testmeasures · A statistical independence test is a hypothesis-testing procedure that evaluates whether variables are statistically independent or conditionally independent based on observed data.

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.