TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

MAR factorization and likelihood ignorability for the full-data parameter

Statement

Assume AC and the hypotheses and notation of Observed-data likelihood under a response model. Suppose the full-data parameter and response-mechanism parameter range independently over the nonempty Cartesian product $Θ \times Ψ$ , so the full-data law is $P_{θ}$ and the response kernel is $g_{ψ}$ . For each pattern let $m_{r}$ denote the pattern reference selected in the lemma: $m_{r} = κ_{r}$ in general, and one may take $m_{r} = ν_{r}^{o b s}$ in the product-reference special case. Let $Q_{θ, r}$ be the law of $W_{o b s} (r)$ under $P_{θ}$ , and choose a finite-valued nonnegative density representative $q_{θ, r} = d Q_{θ, r} / d m_{r}$ . Suppose MAR holds for every $ψ$ and $r$ with one measurable factor $h_{ψ, r}$ valid $P_{θ}$ -almost surely for every $θ$ . Clipping this factor to $[0, 1]$ preserves those almost-sure identities, so take $0 \leq h_{ψ, r} (x) \leq 1$ for every $x$ . Then $ℓ_{θ, ψ} (r, x) = h_{ψ, r} (x) q_{θ, r} (x)$ is a selected observed-record likelihood version relative to the disjoint union reference $λ_{O} (A) = \sum_{r} m_{r} ({x : (r, x) \in A})$ . In the product-reference special case, the missing-coordinate integrals ${\overset{ˉ}{q}}_{θ, r} (x) = \int p_{θ} (x, z) ν_{r}^{m i s} (d z), ℓ_{θ, ψ} (r, x) = \int p_{θ} (x, z) g_{ψ} (r ∣ x, z) ν_{r}^{m i s} (d z)$ are density versions, and may be chosen in the displayed factorized form after selecting finite-valued representatives.

For a fixed response parameter, if $h_{ψ, r} (x) > 0$ and $q_{θ_{2}, r} (x) > 0$ , the factor cancels from the likelihood ratio in $θ$ . Profiling over the same set $Ψ$ for each $θ$ multiplies $q_{θ, r} (x)$ by the factor $\sup_{ψ \in Ψ} h_{ψ, r} (x)$ , which is independent of $θ$ ; when this factor is zero the observed record has zero likelihood for every $θ$ . This licenses omission of the response factor for likelihood inference about $θ$ on its support. It does not license omission when the target depends on $ψ$ , or when the allowed parameter pairs are not a Cartesian product.

Facts & Assumptions

Given: AC; the full-data and response model, nonempty Cartesian parameter space, pattern reference measures, and MAR hypothesis in the Statement.

[F1]

Observed-data likelihood under a response model: the full-data marginal pushforward and each observed pattern sublaw have selected densities relative to the same patternwise reference measures; in the product-reference special case these densities have the missing-coordinate integral versions.

[F2]

MCAR, MAR, and MNAR relative to an observation pattern: MAR means that each pattern probability factors through that pattern's observed projection; for a family of full-data laws the same response mechanism and observed-coordinate function are used across the family.

[F3]

Parametric statistical models: a parameterized model is a map from parameter labels to sampling laws.

[F4]

Law or distribution of a random element: the law under a measurable projection is its pushforward measure; the identity for nonnegative measurable integrands follows from its indicator-set definition by simple approximation and monotone convergence.

[F5]

Every nonnegative measurable function admits an explicit increasing sequence of simple approximations: a nonnegative measurable function is the increasing pointwise limit of nonnegative simple functions.

[F6]

Monotone convergence for the integral: increasing nonnegative measurable functions have integrals increasing to the integral of their pointwise limit.

[F7]

Integrating against a Radon-Nikodym derivative recovers integration against the measure: integration of a simple function against a measure with density $q$ agrees with integration of that function times $q$ against the reference measure; [F5] and [F6] extend this to nonnegative measurable integrands.

[F8]

Likelihood densities: under AC a likelihood density uses a selected finite-valued measurable nonnegative representative of its density class.

[A1]

The Axiom of Choice: AC supplies a choice function for a family of nonempty sets.

Proof

Proof technique: Factor the response probability through the observed projection and use the parameter product.

1.1F2givenalgebra

For each MAR factor in the Statement, replace any measurable almost-sure version $h_{ψ, r}$ by $h_{ψ, r}^{*} = \min (1, \max (0, h_{ψ, r}))$ . Since every kernel evaluation $g_{ψ} (r ∣ w)$ lies in $[0, 1]$ and equals $h_{ψ, r} (π_{r} (w))$ $P_{θ}$ -almost surely for every $θ$ , clipping leaves each of those identities true and makes $h_{ψ, r}^{*}$ pointwise $[0, 1]$ -valued.

2.1A1F1F2F3F4F5F6F7F8step 1.1given

For a measurable observed-coordinate set $B$ , put $u = 1_{B} h_{ψ, r}^{*}$ . By [F4], the integral of any nonnegative simple function of $π_{r} (W)$ under $P_{θ}$ equals its integral under the pushforward law $Q_{θ, r}$ ; [F5] supplies increasing simple approximations to $u$ , and [F6] passes to their limit. Since $Q_{θ, r}$ has density $q_{θ, r}$ with respect to $m_{r}$ , [F7] gives the same integral as $\int_{B} h_{ψ, r}^{*} (x) q_{θ, r} (x) d m_{r} (x)$ , again by simple approximation and monotone convergence. MAR therefore yields $P_{θ, ψ} (R = r, π_{r} (W) \in B) = \int_{B} h_{ψ, r}^{*} (x) q_{θ, r} (x) d m_{r} (x)$ . Thus $h_{ψ, r}^{*} q_{θ, r}$ is a selected density version for the observed pattern section; in the product-reference case this agrees as a density class with the coordinate integral in the Statement.

3.1step 2.1givenalgebra∎

Because $q_{θ, r} (x) \geq 0$ and the same nonempty parameter set $Ψ$ is allowed for every $θ$ , the profile satisfies $\sup_{ψ \in Ψ} ℓ_{θ, ψ} (r, x) = q_{θ, r} (x) \sup_{ψ \in Ψ} h_{ψ, r}^{*} (x)$ . The multiplier is finite and common to every $θ$ , and if it is zero the record has zero likelihood for all $θ$ . For fixed $ψ$ , whenever $h_{ψ, r}^{*} (x) > 0$ and $q_{θ_{2}, r} (x) > 0$ , the ratio at $θ_{1}$ and $θ_{2}$ is $q_{θ_{1}, r} (x) / q_{θ_{2}, r} (x)$ . A target depending on $ψ$ can still depend on the response factor, and a non-Cartesian allowed parameter set need not permit common profiling.

Remarks

The density-class identities are for each fixed parameter pair. As in Likelihood densities, no common exceptional null set across an uncountable parameter family is asserted; the displayed pointwise likelihood claims use the selected versions stated above.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

38 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

Lu Mao, Missing Data: Foundations and Methods, Chapter 1, §1.4 (standard reference, not scraped)
National Research Council, The Prevention and Treatment of Missing Data in Clinical Trials, Chapter 4, pp. 60–61 (standard reference, not scraped)