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.

Randomized procedures as probability kernels

Definition

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment (Statistical experiments) and let $(D, G)$ be a measurable action space: a set of available actions equipped with a sigma-algebra of action events, hence a measurable space (Measurable spaces and measurable sets). A randomized procedure (or randomized decision rule) is a probability kernel $K$ from $(X, A)$ to $(D, G)$ , in the pointwise sense of Measure kernel and probability kernel. Thus $K (x, \cdot)$ is a probability measure on $(D, G)$ for every $x \in X$ , and $x \mapsto K (x, B)$ is $A$ -measurable for every $B \in G$ . After observing $x$ , the procedure assigns probability $K (x, B)$ to taking an action in $B$ . The kernel is specified without reference to the unknown parameter. This definition imposes no measurable structure on $Θ$ . The kernel is the specification of the procedure; no representation by a separate random seed is asserted for arbitrary measurable action spaces.

Depends on

Used by

The one-sample Hotelling F test has exact size alpha Corollary
A nonmeasurable rule is not a statistic or test Counterexample
Bayes risk and minimax risk Definition
Conformal scores and rank p-values Definition
Loss functions and risk Definition
Randomized tests Definition
A coin-flip randomized test Example
A dense Gaussian mean lower bound by Assouad’s method Example
A deterministic statistic is a Dirac kernel Example
A Gaussian two-point lower bound Example
A sparse Gaussian sequence lower bound by Fano’s method Example
Randomization lowers worst-case risk under zero-one loss Example
A randomized procedure has a measurable action law Lemma
A statistic induces a deterministic probability kernel Lemma
Clean block-sparse Gaussian covariance has squared operator lower rate s log(ep/s) over n Lemma
Clean Gaussian covariance has a capped squared operator lower rate Lemma
Every measurable rejection function defines a randomized test Lemma
A posterior-risk minimizer minimizes prior risk Theorem
Assouad lower bound for a squared-error hypercube Theorem
Exact finite-group permutation p-values and tests Theorem
Fano’s packing bound with a reference-distribution KL bound Theorem
Gaussian sparse-sequence minimax lower bound Theorem
Randomization does not improve convex action loss Theorem
Rao–Blackwell theorem for scalar finite-variance estimators Theorem
Two-point Le Cam lower bound for squared-error risk Theorem
Validity of identity-corrected random-permutation Monte Carlo p-values Theorem

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

Ryan Martin, Lecture Notes on Advanced Statistical Theory, §5.1, PDF pp. 100–102 (standard reference, not scraped)
Peter Hoff, Statistical Decision Problems, §§2–4.2, PDF pp. 2–8 (standard reference, not scraped)
Peter Kempthorne, MIT 18.655 Lecture Note 6: Sufficiency, PDF pp. 14–15 (standard reference, not scraped)