# Confidence-interval coverage in a finite model

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Frequentist coverage is the repeated-sampling probability that a fixed interval-valued procedure covers each true parameter.

mko-finite-confidence-interval-coverage · definition · v0.10.0



## Mathematical statement

Supply the complete normalized finite sampling law for every theta. The precommitted procedure I(x)=[L(x),U(x)] is defined at every sample, independent of the unknown theta, and has L(x)≤U(x). Its pointwise coverage is sum_x P_theta(x)*1{L(x)≤theta≤U(x)}; its worst-case coverage is the minimum over the finite theta carrier. An average-only boundary must have distinct pointwise coverages and a target with average≥target>minimum. Posterior probability uses a prior and observed data and is not this coverage. Selection included in the full procedure can be analyzed again; the error is ignoring selection while reusing old coverage. Zero selected mass makes the corresponding conditional coverage undefined.





## Representative formula (exact TeX)

I(T,x,t)=c





## Assumptions


- Theta and sample carriers are finite and unique, with exact normalized laws for every theta.
- The total interval procedure has ordered closed endpoints.
- Any selection rule must enter the procedure before coverage is recalculated.





## Symbol scope


- I — confidence-interval procedure
- T — finite parameter set
- x — sample outcome
- t — true parameter
- c — coverage probability





## Intuition, definition, and misconceptions

Qualify complete laws and a total precommitted procedure first.

Calculate every pointwise coverage and its worst-case minimum.

Separate weighted averages, posterior probabilities and selection-adjusted procedures.

95% coverage is not a95% posterior probability for theta after observing a sample; selection cannot be ignored.





## Adopted derivation guide

[Pointwise coverage, averaging and selection](https://ocme.evemisslab.com/en/derivations/derivation-mko-finite-confidence-interval-coverage-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Sampling distribution](https://ocme.evemisslab.com/en/mko/mko-sampling-distribution/) — Use the complete sampling law.
- [Statistical estimator](https://ocme.evemisslab.com/en/mko/mko-statistical-estimator/) — Retain the data-dependent procedure background.
- [Parametric statistical model](https://ocme.evemisslab.com/en/mko/mko-parametric-statistical-model/) — Keep the theta-indexed laws.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — Calculate coverage probabilities.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Test closed-interval membership.





## Verification boundary

Finite coverage does not imply asymptotic validity or a posterior claim for one dataset. A separate synthetic diagnostic records a frozen-helper lexical/numeric theta-order gap; the current questions all use[0,1] and remain unaffected. No source repair is adopted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add test inversion, randomized intervals and selective inference.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/finite-confidence-interval-coverage.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-finite-confidence-interval-coverage/index.md)