Confidence-interval coverage in a finite model

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Sampling distribution — Use the complete sampling law.
• Statistical estimator — Retain the data-dependent procedure background.
• Parametric statistical model — Keep the theta-indexed laws.
• Probability on finite sample spaces — Calculate coverage probabilities.
• Usual order on the real numbers — 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.