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.

Observational equivalence

Definition

In a parametric model $P : Θ \to P (X, A)$ , two parameter values $θ, η \in Θ$ are observationally equivalent, written $θ \sim η$ , if and only if

P_{θ} = P_{η},

that is, $P_{θ} (A) = P_{η} (A)$ for every $A \in A$ (Parametric statistical models). Equivalent parameter values therefore give exactly the same sampling law. No distinctness of $θ$ and $η$ is required.

Depends on

Parametric statistical models

Used by

A full-rank local Hessian does not capture both posterior modes Counterexample
A Gaussian synthetic likelihood can change an exact posterior Counterexample
A nonconstant summary can lose an identifiable parameter Counterexample
An identifiable model can have persistent ABC summary distortion Counterexample
The observed first time and cause do not identify the latent joint failure-time law Counterexample
Identifiability of an estimand Definition
Positivity failure, near-positivity weights, and an MNAR sensitivity curve Example
The same Bernoulli law with two different targets Example
The same observed law can have different intervention laws Example
Backdoor adjustment identifies an interventional law Theorem
Exact binary-outcome bounds under unrestricted missingness Theorem
Front-door adjustment identifies an interventional mean Theorem
Identifiability is constancy on observational classes Theorem
Nonidentifiability precludes simultaneous consistency Theorem

Dependency tree · two levels

2 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

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §§1.3.6 and 1.4.1, PDF pp. 20–23 (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families, §6, PDF pp. 6–7 (standard reference, not scraped)