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.

Identifiability of an estimand

Definition

An estimand $ψ : Θ \to T_{0}$ is identifiable in a parametric model $P$ when

P_{θ} = P_{η} ⟹ ψ (θ) = ψ (η) (θ, η \in Θ),

or equivalently when $ψ$ is constant on each set of parameter values with the same sampling law (Parametric statistical models, Observational equivalence, Estimands on a statistical model). The model is identifiable when its identity target ${id}_{Θ} : Θ \to Θ$ is identifiable, equivalently when $P$ is injective.

Depends on

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
A redundant Bernoulli parameterization has a nonidentifiable coordinate Counterexample
An identifiable model can have persistent ABC summary distortion Counterexample
Core IV conditions alone do not identify the population ATE Counterexample
Identifiability and KL support do not replace testability Counterexample
Single-world assumptions do not identify natural effects Counterexample
The observed first time and cause do not identify the latent joint failure-time law Counterexample
Binding function and indirect-inference target Definition
Scientific statistical model contracts Definition
A finite identifiable model with posterior consistency Example
A non-injective binding map cannot recover a sign Example
A two-stratum standardization and IPW calculation Example
An unvalidated calibration offset leaves the true quantity unidentified Example
Extreme IPW weights under poor overlap Example
Identifiable Bernoulli model and parameter target Example
Positivity failure leaves the target unsupported Example
Positivity failure prevents standardization and IPW identification Example
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
Weight truncation changes the IPW target Example
The observed law need not identify the latent-error variance split Proposition
A constant additive effect is identified by the Wald ratio Theorem
A zero inclusion probability prevents total identification and universal unbiased estimation Theorem
Backdoor adjustment identifies an interventional law Theorem
Conditional-class standardization identifies a causal mean Theorem
Discrete-treatment potential-outcome means are identifiable Theorem
Exact binary-outcome bounds under unrestricted missingness Theorem
Front-door adjustment identifies an interventional mean Theorem
Gaussian variance-component identifiability is covariance-map injectivity Theorem
Identifiability is constancy on observational classes Theorem
Inverse probability weighting identifies a causal mean Theorem
Nonidentifiability precludes simultaneous consistency Theorem
The natural-effect mediation formula under independent errors Theorem
The Wald ratio identifies the complier average treatment effect Theorem

Dependency tree · two levels

4 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, §§1.3.5–1.3.6, PDF pp. 19–21 (standard reference, not scraped)
Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families, §6, PDF pp. 6–7 (standard reference, not scraped)