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.
Gaussian branch of the linear mixed model
Definition
The Gaussian branch of Finite-dimensional linear mixed model with explicit longitudinal roles adds the following assumption: under the Axiom of Choice, the concatenated vector has the multivariate-normal law of Multivariate normal law, including singular covariance. Thus the full latent-effect and residual vector is jointly Gaussian. This is an additional branch; the moment-only definition and the covariance identity do not assume Gaussianity.
Remarks
This definition carries AC because the current multivariate-normal supplier does. Singular or is permitted by that supplier; any later inverse or density requires a separate positive-definiteness hypothesis.
Depends on
Used by
- Two visits can alias random-slope and residual variance components Counterexample
- Restricted maximum likelihood from error contrasts Definition
- A Gaussian error contrast has the induced multivariate-normal law Lemma
- Conditional Gaussian law of random effects Theorem
- Gaussian variance-component identifiability is covariance-map injectivity Theorem
Dependency tree · two levels
20 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
- Fan Li, STA610 Lecture 6: Linear Mixed-Effects Models, PDF pp. 4–5 (standard reference, not scraped)
- Duchateau, Janssen, and Rowlands, Linear Mixed Models, Chapter 2 §§2.3.3–2.4.2 (standard reference, not scraped)