AlgorithmAlgorithm: 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.

Bulk effective sample size

Definition

Inputs: $M \geq 2$ equal-length scalar chains with $N \geq 2$ observations each. Output: A finite bulk effective-sample-size estimate or not estimable.

If $N < 4$ , return not estimable. This procedure calls Split multi-chain autocorrelation effective sample size, whose input chains must have length at least $4$ .
Pool all draws and apply Blom rank-normal scores for pooled draws. Apply Split multi-chain autocorrelation effective sample size to the resulting $M$ transformed chains and return its output.

Rank normalization gives finite transformed values for every finite real-valued draw array, including arrays whose target distribution might have infinite mean or variance. It is an efficiency diagnostic for the transformed values and is not, by itself, a Monte Carlo standard error for the original posterior mean.

Depends on

Used by

Centered eight-schools runs show small R-hat with low tail ESS Example

Dependency tree · two levels

8 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

Aki Vehtari et al., Rank-normalization, folding, and localization, §4.1 (standard reference, not scraped)
posterior package reference, Bulk effective sample size (standard reference, not scraped)