Alphabeta MathHow 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.
Conditional exchangeability of potential outcomes
Definition
Let be the observed treatment assignment, a baseline covariate random element, and the potential-outcome family of Potential outcomes and consistency. Conditional exchangeability given means
where conditional independence is the bounded-measurable-function factorization of Conditional independence given a sigma-algebra under the The Axiom of Choice. Each fixed is then conditionally independent of given by measurable-map preservation in Conditional-independence equivalences and preservation. This is a full-data assumption; the observed distribution alone does not establish it.
Depends on
Used by
- Binary confounding and standardization Example
- Positivity failure leaves the target unsupported Example
- Randomized treatment identifies the target mean contrast Example
- Conditional-class standardization identifies a causal mean Theorem
- Discrete-treatment potential-outcome means are identifiable Theorem
- Inverse probability weighting identifies a causal mean Theorem
Dependency tree · two levels
12 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)