AlgorithmAlgorithm: Literature-sourcedProof: Not applicablePipeline-generated

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.

Tail effective sample size

Definition

Inputs: $M \geq 2$ equal-length chains of scalar draws with $N \geq 2$ observations each. Output: The minimum of the two tail ESS estimates 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 the draws and calculate $q_{.05} = Q_{S} (.05)$ and $q_{.95} = Q_{S} (.95)$ using Left empirical quantile of finite draws. For each $p \in {.05, .95}$ , form the bounded indicator chains $I_{m, n}^{(p)} = 1 {x_{m, n} \leq q_{p}}$ , pool their values, apply Blom rank-normal scores for pooled draws, and apply Split multi-chain autocorrelation effective sample size to the transformed indicator chains.
If either indicator ESS is not estimable, return not estimable. Otherwise return the minimum of the two values.

This is a tail-focused efficiency diagnostic; it does not establish that a quantile estimate has a specified error bound.

Depends on

Used by

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

Dependency tree · two levels

9 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, §4.3 (standard reference, not scraped)