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.
Statistical experiments
Definition
An experiment is a measurable space , a set of parameter labels, and a family of probability measures on (Measurable spaces and measurable sets, Probability measures and probability spaces). The measures are the sampling laws. If data under a parameter value are realized as a random element, its law is the pushforward measure from Law or distribution of a random element; it is a probability measure by The law of a random element is a probability measure.
The definition specifies the family of laws only. It supplies no sigma-algebra on , no measurability condition on , and no common underlying probability space on which all observations must be realized.
Depends on
Used by
- Exact selective tests and confidence sets from the pivot Corollary
- A full-rank local Hessian does not capture both posterior modes Counterexample
- A Gaussian synthetic likelihood can change an exact posterior Counterexample
- A generalized Bayes credible set can have zero coverage Counterexample
- A measurable [0,1]-valued statistic need not be a valid p-value Counterexample
- A nonconstant summary can lose an identifiable parameter Counterexample
- A nonmeasurable rule is not a statistic or test Counterexample
- A redundant Bernoulli parameterization has a nonidentifiable coordinate Counterexample
- Finite power-oracle queries cannot universally certify no feasible sample size Counterexample
- Positive covariance does not imply PRDS Counterexample
- Adversarial sample replacement Definition
- Bayesian model as a prior and sampling kernel Definition
- Confidence sets and coverage Definition
- Contamination-robust expected risk and tail risk Definition
- Critical, detection, and quantification levels Definition
- Density targets and the kernel density estimator Definition
- Dominated statistical experiments Definition
- Estimator sequences in IID experiments Definition
- Exact selective confidence set for a parameter functional Definition
- Families with a monotone likelihood ratio Definition
- Finite populations and sampling designs Definition
- Finite randomized treatment-assignment models Definition
- Fixed-design linear regression model Definition
- Huber contamination model Definition
- Huber contamination modulus Definition
- Likelihood-ratio tests with boundary randomization Definition
- Loss functions and risk Definition
- Marginal likelihoods, proper model evidence, and Bayes factors Definition
- Measurable finite-group permutation setups Definition
- Null-valid p-values and familywise and false-discovery error rates Definition
- Parameter-measurable statistical models as kernels Definition
- Parametric statistical models Definition
- Pivotal quantities Definition
- Power and sample-size planning specifications Definition
- Randomized procedures as probability kernels Definition
- Randomized tests Definition
- Sampling units and the resampling-unit contract Definition
- Scientific statistical model contracts Definition
- Statistics and estimators Definition
- Test size and power Definition
…and 43 more results.
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
- Ryan Martin, Lecture Notes on Advanced Statistical Theory, §§1.3.6 and 1.4.1, PDF pp. 20–23 (standard reference, not scraped)
- Charles J. Geyer, Stat 5421 Lecture Notes: Exponential Families, §6, PDF pp. 6–7 (standard reference, not scraped)