DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Average and conditional average treatment effects

Definition

In a fixed target population with two specified treatment versions $0$ and $1$ and integrable potential outcomes $Y^{0}, Y^{1}$ from Potential outcomes and consistency, the average treatment effect (ATE) is the target-population mean contrast $E [Y^{1} - Y^{0}]$ . For a baseline covariate random element $L$ , the conditional average treatment effect (CATE) given $L$ is the conditional-expectation almost-sure class

$E [Y^{1} - Y^{0} ∣ σ (L)],$

using the conditional-expectation convention of Conditional expectation as an ae class under the The Axiom of Choice. Thus a CATE is determined only up to equality almost surely under the target law of $L$ ; a pointwise value at a null covariate value is not specified by this definition. The ATE is the mean of the individual contrast, not an assertion that both potential outcomes are observed for any unit.

Depends on

Used by

Controlled and natural mediation effects Definition
Binary confounding and standardization Example
Positivity failure leaves the target unsupported Example
Randomized treatment identifies the target mean contrast Example
Discrete-treatment potential-outcome means are identifiable Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Miguel A. Hernán and James M. Robins, Causal Inference: What If, 2020 (author revision 19 August 2026) (standard reference, not scraped)
Hyunseung Kang, Causal Inference: Identification Under Strong Ignorability, UW–Madison STAT 992 lecture note (standard reference, not scraped)