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.

Loss functions and risk

Definition

Let $E = ((X, A), Θ, (P_{θ})_{θ \in Θ})$ be a statistical experiment (Statistical experiments), let $(D, G)$ be a measurable action space, meaning a set with a specified sigma-algebra (Measurable spaces and measurable sets), and let $L : Θ \times D \to [0, + \infty]$ be a loss function whose section $a \mapsto L (θ, a)$ is $G$ -measurable for each $θ$ . A randomized decision rule is a probability kernel $δ$ from the data space to the action space (Randomized procedures as probability kernels). Its risk at $θ$ is the extended nonnegative number $R (θ, δ) = \int_{X} (\int_{D} L (θ, a) δ (x, d a)) P_{θ} (d x) .$ The inner integral is a measurable function of $x$ , by A randomized procedure has a measurable action law, so the outer nonnegative integral is defined by Expectation of a nonnegative or integrable random variable. A deterministic measurable rule is included by taking its point-mass action kernel. No measurability on $Θ$ is required for this definition.

Depends on

Used by

Matching minimax rate in a Gaussian orthogonal sequence model Corollary
An open action set need not contain a posterior-risk minimizer Counterexample
Collinear columns identify a fitted mean, not its coefficients Counterexample
Bayes risk and minimax risk Definition
Bias and mean squared error Definition
Contamination-robust expected risk and tail risk Definition
Cross-validation splits and training-size risk targets Definition
Finite posterior risk and Bayes action rule Definition
Huber contamination modulus Definition
Pointwise squared-error risk for regression Definition
Risk domination and admissibility Definition
Test size and power Definition
A data-dependent rule dominates a constant rule Example
A dense Gaussian mean lower bound by Assouad’s method Example
A Gaussian two-point lower bound Example
A sparse Gaussian sequence lower bound by Fano’s method Example
Constrained least squares in a two-coordinate orthogonal sequence model Example
Randomization lowers worst-case risk under zero-one loss Example
Squared-error risk for a Bernoulli observation Example
Zero training residual and positive test error Example
Binary zero-one loss risks are the testing error probabilities Lemma
A posterior-risk minimizer minimizes prior risk Theorem
Assouad lower bound for a squared-error hypercube Theorem
BIC fitted-value risk bound Theorem
Exact squared-error risk of an isotropic projection fit Theorem
Fano’s packing bound with a reference-distribution KL bound Theorem
Gaussian sparse-sequence minimax lower bound Theorem
Huber contamination modulus lower bound for tail risk Theorem
Local asymptotic minimax bound Theorem
Randomization does not improve convex action loss Theorem
Sparse constrained least-squares risk bound Theorem
Sub-Gaussian projection risk bounds Theorem
Two-point Le Cam lower bound for squared-error risk Theorem

Dependency tree · two levels

23 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, §§4.1–4.2, PDF pp. 7–8 (standard reference, not scraped)