← Return to results
Precommitted Finite Intervals, Pointwise Coverage and Selection Boundaries
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[1/3, 2/3], [2/3, 1/3]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[1/3, 2/3], [2/3, 1/3]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[1/3, 2/3], [2/3, 1/3]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
2. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[1/2, 1/2], [1/2, 1/2]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[1/2, 1/2], [1/2, 1/2]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
3. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[3/5, 2/5], [2/5, 3/5]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[3/5, 2/5], [2/5, 3/5]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[3/5, 2/5], [2/5, 3/5]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
4. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[4/6, 2/6], [2/6, 4/6]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[2/3, 1/3], [1/3, 2/3]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[2/3, 1/3], [1/3, 2/3]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
5. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[5/7, 2/7], [2/7, 5/7]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[5/7, 2/7], [2/7, 5/7]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[5/7, 2/7], [2/7, 5/7]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
6. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[3/4, 1/4], [1/4, 3/4]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[3/4, 1/4], [1/4, 3/4]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
7. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[7/9, 2/9], [2/9, 7/9]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[7/9, 2/9], [2/9, 7/9]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[7/9, 2/9], [2/9, 7/9]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
8. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[8/10, 2/10], [2/10, 8/10]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[4/5, 1/5], [1/5, 4/5]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[4/5, 1/5], [1/5, 4/5]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
9. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[9/11, 2/11], [2/11, 9/11]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[9/11, 2/11], [2/11, 9/11]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[9/11, 2/11], [2/11, 9/11]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
10. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[10/12, 2/12], [2/12, 10/12]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[5/6, 1/6], [1/6, 5/6]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[5/6, 1/6], [1/6, 5/6]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
11. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[11/13, 2/13], [2/13, 11/13]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[11/13, 2/13], [2/13, 11/13]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[11/13, 2/13], [2/13, 11/13]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
12. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[12/14, 2/14], [2/14, 12/14]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[6/7, 1/7], [1/7, 6/7]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[6/7, 1/7], [1/7, 6/7]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
13. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[13/15, 2/15], [2/15, 13/15]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[13/15, 2/15], [2/15, 13/15]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[13/15, 2/15], [2/15, 13/15]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
14. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
15. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[15/17, 2/17], [2/17, 15/17]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[15/17, 2/17], [2/17, 15/17]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[15/17, 2/17], [2/17, 15/17]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
16. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[16/18, 2/18], [2/18, 16/18]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[8/9, 1/9], [1/9, 8/9]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[8/9, 1/9], [1/9, 8/9]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
17. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[17/19, 2/19], [2/19, 17/19]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[17/19, 2/19], [2/19, 17/19]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[17/19, 2/19], [2/19, 17/19]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
18. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[18/20, 2/20], [2/20, 18/20]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[9/10, 1/10], [1/10, 9/10]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[9/10, 1/10], [1/10, 9/10]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
19. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[19/21, 2/21], [2/21, 19/21]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[19/21, 2/21], [2/21, 19/21]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[19/21, 2/21], [2/21, 19/21]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
20. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[20/22, 2/22], [2/22, 20/22]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[10/11, 1/11], [1/11, 10/11]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[10/11, 1/11], [1/11, 10/11]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
21. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[21/23, 2/23], [2/23, 21/23]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[21/23, 2/23], [2/23, 21/23]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[21/23, 2/23], [2/23, 21/23]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
22. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[22/24, 2/24], [2/24, 22/24]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[11/12, 1/12], [1/12, 11/12]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[11/12, 1/12], [1/12, 11/12]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
23. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[23/25, 2/25], [2/25, 23/25]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[23/25, 2/25], [2/25, 23/25]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[23/25, 2/25], [2/25, 23/25]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
24. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[24/26, 2/26], [2/26, 24/26]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[12/13, 1/13], [1/13, 12/13]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[12/13, 1/13], [1/13, 12/13]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
25. A Complete Precommitted Interval Procedure Keeps Every Probability Alignment
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[25/27, 2/27], [2/27, 25/27]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (theta_values, samples, aligned_closed_intervals, aligned_probability_rows, procedure_qualified, qualification_statement).
Intervals align with samples; probability rows align with theta and samples. The procedure is total, precommitted and independent of the unknown theta. A separate unadopted frozen-helper sorting diagnostic concerns cross-digit theta values; all current rows use[0,1].
View answer and explanation
([0, 1], [a, b], [[0, 0], [1, 1]], [[25/27, 2/27], [2/27, 25/27]], Yes, total_precommitted_theta_independent_interval_procedure)
Qualify all closed endpoints and the entire normalized law for each theta. Preserve sample/interval and theta/row associations under canonical ordering. The procedure may depend on data but not on the unknown true theta. This is finite-model qualification, not an asymptotic coverage theorem. The exact result, retaining the requested order, is ([0, 1], [a, b], [[0, 0], [1, 1]], [[25/27, 2/27], [2/27, 25/27]], Yes, total_precommitted_theta_independent_interval_procedure).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-procedure-qualification · candidate
26. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[1/3, 2/3], [2/3, 1/3]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 1/3], [1, 1/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 1/3], [1, 1/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
27. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 1/2], [1, 1/2]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
28. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[3/5, 2/5], [2/5, 3/5]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 3/5], [1, 3/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 3/5], [1, 3/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
29. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[4/6, 2/6], [2/6, 4/6]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 2/3], [1, 2/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 2/3], [1, 2/3]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
30. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[5/7, 2/7], [2/7, 5/7]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 5/7], [1, 5/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 5/7], [1, 5/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
31. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 3/4], [1, 3/4]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 3/4], [1, 3/4]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
32. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[7/9, 2/9], [2/9, 7/9]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 7/9], [1, 7/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 7/9], [1, 7/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
33. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[8/10, 2/10], [2/10, 8/10]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 4/5], [1, 4/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 4/5], [1, 4/5]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
34. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[9/11, 2/11], [2/11, 9/11]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 9/11], [1, 9/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 9/11], [1, 9/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
35. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[10/12, 2/12], [2/12, 10/12]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 5/6], [1, 5/6]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 5/6], [1, 5/6]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
36. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[11/13, 2/13], [2/13, 11/13]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 11/13], [1, 11/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 11/13], [1, 11/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
37. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[12/14, 2/14], [2/14, 12/14]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 6/7], [1, 6/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 6/7], [1, 6/7]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
38. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[13/15, 2/15], [2/15, 13/15]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 13/15], [1, 13/15]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 13/15], [1, 13/15]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
39. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 7/8], [1, 7/8]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 7/8], [1, 7/8]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
40. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[15/17, 2/17], [2/17, 15/17]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 15/17], [1, 15/17]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 15/17], [1, 15/17]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
41. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[16/18, 2/18], [2/18, 16/18]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 8/9], [1, 8/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 8/9], [1, 8/9]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
42. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[17/19, 2/19], [2/19, 17/19]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 17/19], [1, 17/19]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 17/19], [1, 17/19]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
43. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[18/20, 2/20], [2/20, 18/20]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 9/10], [1, 9/10]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 9/10], [1, 9/10]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
44. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[19/21, 2/21], [2/21, 19/21]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 19/21], [1, 19/21]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 19/21], [1, 19/21]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
45. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[20/22, 2/22], [2/22, 20/22]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 10/11], [1, 10/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 10/11], [1, 10/11]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
46. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[21/23, 2/23], [2/23, 21/23]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 21/23], [1, 21/23]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 21/23], [1, 21/23]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
47. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[22/24, 2/24], [2/24, 22/24]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 11/12], [1, 11/12]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 11/12], [1, 11/12]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
48. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[23/25, 2/25], [2/25, 23/25]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 23/25], [1, 23/25]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 23/25], [1, 23/25]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
49. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[24/26, 2/26], [2/26, 24/26]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 12/13], [1, 12/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 12/13], [1, 12/13]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
50. Coverage Is a Repeated-Sampling Probability at Each Fixed Theta
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[25/27, 2/27], [2/27, 25/27]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. Give (pointwise_coverage_table, coverage_definition, frequentist_scope).
View answer and explanation
([[0, 25/27], [1, 25/27]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability)
For each fixed theta sum exactly P_theta(x) over all samples whose reported interval contains theta, including endpoints. Keep the full pointwise table. This is not posterior probability for theta after observing one sample. The exact result, retaining the requested order, is ([[0, 25/27], [1, 25/27]], coverage_theta=P_theta(theta_in_I(X)), not_a_posterior_probability).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-pointwise-coverage · candidate
51. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[1/3, 2/3], [2/3, 1/3]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 1/3], [1, 1/3]], 1/3, 1/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 1/3], [1, 1/3]], 1/3, 1/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
52. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[2/4, 2/4], [2/4, 2/4]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 1/2], [1, 1/2]], 1/2, 1/2, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 1/2], [1, 1/2]], 1/2, 1/2, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
53. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[3/5, 2/5], [2/5, 3/5]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 3/5], [1, 3/5]], 3/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 3/5], [1, 3/5]], 3/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
54. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[4/6, 2/6], [2/6, 4/6]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 2/3], [1, 2/3]], 2/3, 2/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 2/3], [1, 2/3]], 2/3, 2/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
55. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[5/7, 2/7], [2/7, 5/7]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 5/7], [1, 5/7]], 5/7, 5/7, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 5/7], [1, 5/7]], 5/7, 5/7, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
56. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[6/8, 2/8], [2/8, 6/8]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 3/4], [1, 3/4]], 3/4, 1/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 3/4], [1, 3/4]], 3/4, 1/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
57. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[7/9, 2/9], [2/9, 7/9]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 7/9], [1, 7/9]], 7/9, 1/2, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 7/9], [1, 7/9]], 7/9, 1/2, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
58. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[8/10, 2/10], [2/10, 8/10]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 4/5], [1, 4/5]], 4/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 4/5], [1, 4/5]], 4/5, 3/5, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
59. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[9/11, 2/11], [2/11, 9/11]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 9/11], [1, 9/11]], 9/11, 2/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 9/11], [1, 9/11]], 9/11, 2/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
60. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[10/12, 2/12], [2/12, 10/12]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 5/6], [1, 5/6]], 5/6, 5/7, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 5/6], [1, 5/6]], 5/6, 5/7, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
61. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[11/13, 2/13], [2/13, 11/13]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 11/13], [1, 11/13]], 11/13, 1/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 11/13], [1, 11/13]], 11/13, 1/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
62. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[12/14, 2/14], [2/14, 12/14]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 6/7], [1, 6/7]], 6/7, 1/2, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 6/7], [1, 6/7]], 6/7, 1/2, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
63. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[13/15, 2/15], [2/15, 13/15]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 13/15], [1, 13/15]], 13/15, 3/5, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 13/15], [1, 13/15]], 13/15, 3/5, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
64. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[14/16, 2/16], [2/16, 14/16]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 7/8], [1, 7/8]], 7/8, 2/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 7/8], [1, 7/8]], 7/8, 2/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
65. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[15/17, 2/17], [2/17, 15/17]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 15/17], [1, 15/17]], 15/17, 5/7, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 15/17], [1, 15/17]], 15/17, 5/7, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
66. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[16/18, 2/18], [2/18, 16/18]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 8/9], [1, 8/9]], 8/9, 1/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 8/9], [1, 8/9]], 8/9, 1/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
67. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[17/19, 2/19], [2/19, 17/19]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 17/19], [1, 17/19]], 17/19, 1/2, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 17/19], [1, 17/19]], 17/19, 1/2, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
68. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[18/20, 2/20], [2/20, 18/20]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 9/10], [1, 9/10]], 9/10, 3/5, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 9/10], [1, 9/10]], 9/10, 3/5, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
69. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[19/21, 2/21], [2/21, 19/21]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 19/21], [1, 19/21]], 19/21, 2/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 19/21], [1, 19/21]], 19/21, 2/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
70. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[20/22, 2/22], [2/22, 20/22]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 10/11], [1, 10/11]], 10/11, 5/7, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 10/11], [1, 10/11]], 10/11, 5/7, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
71. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[21/23, 2/23], [2/23, 21/23]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=1/3. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 21/23], [1, 21/23]], 21/23, 1/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 21/23], [1, 21/23]], 21/23, 1/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
72. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[22/24, 2/24], [2/24, 22/24]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=2/4. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 11/12], [1, 11/12]], 11/12, 1/2, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 11/12], [1, 11/12]], 11/12, 1/2, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
73. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[23/25, 2/25], [2/25, 23/25]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=3/5. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 23/25], [1, 23/25]], 23/25, 3/5, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 23/25], [1, 23/25]], 23/25, 3/5, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
74. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[24/26, 2/26], [2/26, 24/26]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=4/6. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 12/13], [1, 12/13]], 12/13, 2/3, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 12/13], [1, 12/13]], 12/13, 2/3, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
75. A Uniform Coverage Target Is Checked against the Minimum, Not an Average
Use complete theta values=[0, 1], samples=[a, b], theta/sample-aligned probability rows=[[25/27, 2/27], [2/27, 25/27]], and sample-aligned closed intervals=[[0, 0], [1, 1]]. The interval procedure is total, precommitted and independent of the unknown theta. The declared target is=5/7. Give (pointwise_coverage_table, minimum_coverage, target, uniform_target_met, minimum_definition).
View answer and explanation
([[0, 25/27], [1, 25/27]], 25/27, 5/7, Yes, worst_case_is_minimum_over_finite_theta)
Calculate every pointwise coverage and take the minimum over the finite theta carrier. Compare that exact minimum with the target. Weighted average coverage cannot substitute for this uniform criterion. The exact result, retaining the requested order, is ([[0, 25/27], [1, 25/27]], 25/27, 5/7, Yes, worst_case_is_minimum_over_finite_theta).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-worst-case-coverage · candidate
76. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[1,3],[2,3]],[[2,3],[1,3]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
77. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=posterior_confusion and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[2,4],[2,4]],[[2,4],[2,4]]],"intervals":[[0,0],[1,1]],"prior":[[2,4],[2,4]],"observed":"b"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(posterior_confusion, b, [1/2, 1/2], [[0, 1/2], [1, 1/2]], No, posterior_given_observed_data_is_not_frequentist_coverage)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/2, 1/2], [[0, 1/2], [1, 1/2]], No, posterior_given_observed_data_is_not_frequentist_coverage).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
78. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=ignored_selection and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[3,5],[2,5]],[[2,5],[3,5]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(ignored_selection, [a], [[0, 3/5], [1, 3/5]], [[0, 3/5, 1], [1, 2/5, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 3/5], [1, 3/5]], [[0, 3/5, 1], [1, 2/5, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
79. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=invalid_procedure and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[4,6],[2,6]],[[2,6],[4,6]]],"intervals":[[4,3],[4,4]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
80. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[5,7],[2,7]],[[2,7],[5,7]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
81. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=posterior_confusion and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[6,8],[2,8]],[[2,8],[6,8]]],"intervals":[[0,0],[1,1]],"prior":[[2,4],[2,4]],"observed":"b"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(posterior_confusion, b, [1/4, 3/4], [[0, 3/4], [1, 3/4]], No, posterior_given_observed_data_is_not_frequentist_coverage)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/4, 3/4], [[0, 3/4], [1, 3/4]], No, posterior_given_observed_data_is_not_frequentist_coverage).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
82. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=ignored_selection and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[7,9],[2,9]],[[2,9],[7,9]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(ignored_selection, [a], [[0, 7/9], [1, 7/9]], [[0, 7/9, 1], [1, 2/9, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 7/9], [1, 7/9]], [[0, 7/9, 1], [1, 2/9, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
83. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=invalid_procedure and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[8,10],[2,10]],[[2,10],[8,10]]],"intervals":[[8,7],[8,8]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
84. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[9,11],[2,11]],[[2,11],[9,11]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
85. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=posterior_confusion and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[10,12],[2,12]],[[2,12],[10,12]]],"intervals":[[0,0],[1,1]],"prior":[[2,4],[2,4]],"observed":"b"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(posterior_confusion, b, [1/6, 5/6], [[0, 5/6], [1, 5/6]], No, posterior_given_observed_data_is_not_frequentist_coverage)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/6, 5/6], [[0, 5/6], [1, 5/6]], No, posterior_given_observed_data_is_not_frequentist_coverage).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
86. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=ignored_selection and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[11,13],[2,13]],[[2,13],[11,13]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(ignored_selection, [a], [[0, 11/13], [1, 11/13]], [[0, 11/13, 1], [1, 2/13, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 11/13], [1, 11/13]], [[0, 11/13, 1], [1, 2/13, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
87. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=invalid_procedure and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[12,14],[2,14]],[[2,14],[12,14]]],"intervals":[[12,11],[12,12]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
88. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[13,15],[2,15]],[[2,15],[13,15]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
89. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=posterior_confusion and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[14,16],[2,16]],[[2,16],[14,16]]],"intervals":[[0,0],[1,1]],"prior":[[2,4],[2,4]],"observed":"b"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(posterior_confusion, b, [1/8, 7/8], [[0, 7/8], [1, 7/8]], No, posterior_given_observed_data_is_not_frequentist_coverage)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/8, 7/8], [[0, 7/8], [1, 7/8]], No, posterior_given_observed_data_is_not_frequentist_coverage).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
90. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=ignored_selection and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[15,17],[2,17]],[[2,17],[15,17]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(ignored_selection, [a], [[0, 15/17], [1, 15/17]], [[0, 15/17, 1], [1, 2/17, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 15/17], [1, 15/17]], [[0, 15/17, 1], [1, 2/17, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
91. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=invalid_procedure and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[16,18],[2,18]],[[2,18],[16,18]]],"intervals":[[16,15],[16,16]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
92. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[17,19],[2,19]],[[2,19],[17,19]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
93. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=posterior_confusion and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[18,20],[2,20]],[[2,20],[18,20]]],"intervals":[[0,0],[1,1]],"prior":[[2,4],[2,4]],"observed":"b"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(posterior_confusion, b, [1/10, 9/10], [[0, 9/10], [1, 9/10]], No, posterior_given_observed_data_is_not_frequentist_coverage)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/10, 9/10], [[0, 9/10], [1, 9/10]], No, posterior_given_observed_data_is_not_frequentist_coverage).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
94. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=ignored_selection and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[19,21],[2,21]],[[2,21],[19,21]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(ignored_selection, [a], [[0, 19/21], [1, 19/21]], [[0, 19/21, 1], [1, 2/21, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 19/21], [1, 19/21]], [[0, 19/21, 1], [1, 2/21, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
95. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=invalid_procedure and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[20,22],[2,22]],[[2,22],[20,22]]],"intervals":[[20,19],[20,20]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
96. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[21,23],[2,23]],[[2,23],[21,23]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
97. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=posterior_confusion and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[22,24],[2,24]],[[2,24],[22,24]]],"intervals":[[0,0],[1,1]],"prior":[[2,4],[2,4]],"observed":"b"}. Give (mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(posterior_confusion, b, [1/12, 11/12], [[0, 11/12], [1, 11/12]], No, posterior_given_observed_data_is_not_frequentist_coverage)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (posterior_confusion, b, [1/12, 11/12], [[0, 11/12], [1, 11/12]], No, posterior_given_observed_data_is_not_frequentist_coverage).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
98. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=ignored_selection and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[23,25],[2,25]],[[2,25],[23,25]]],"intervals":[[0,0],[1,1]],"selected_samples":["a"]}. Give (mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(ignored_selection, [a], [[0, 23/25], [1, 23/25]], [[0, 23/25, 1], [1, 2/25, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (ignored_selection, [a], [[0, 23/25], [1, 23/25]], [[0, 23/25, 1], [1, 2/25, 0]], No, selection_must_be_included_and_coverage_recomputed_not_declared_automatically_invalid).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
99. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=invalid_procedure and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[24,26],[2,26]],[[2,26],[24,26]]],"intervals":[[24,23],[24,24]]}. Give (mode, procedure_valid, validation_error, procedure_qualification_claim).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (invalid_procedure, No, interval lower endpoint exceeds upper endpoint, No).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
100. Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities
Use mode=average_only and complete model={"theta_values":[0,1],"samples":["a","b"],"probability_rows":[[[25,27],[2,27]],[[2,27],[25,27]]],"intervals":[[0,0],[0,0]],"theta_weights":[[1,3],[2,3]],"target":[1,3]}. Give (mode, pointwise_coverage_table, theta_weights, weighted_average, minimum_coverage, target, distinct_coverages, average_strictly_above_minimum, average_masks_target_failure, average_implies_uniform_target_claim, boundary).
The four modes have11/6/6/4 fields. Conditional coverage is undefined at zero selected mass. Selection is not automatically invalid: the error is reusing coverage without accounting for the selection rule.
View answer and explanation
(average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target)
average_only requires genuinely distinct coverages with weighted average≥target>minimum. posterior_confusion uses the explicit prior and observed sample to compute a different conditional quantity. ignored_selection retains each selected probability mass and recomputed conditional coverage, using undefined at zero mass. Selection is not automatically invalid; ignoring it is the error. invalid_procedure retains the actual validation error. The exact result, retaining the requested order, is (average_only, [[0, 1], [1, 0]], [1/3, 2/3], 1/3, 0, 1/3, Yes, Yes, Yes, No, average_coverage_can_mask_failure_of_uniform_target).
Reviewed step-by-step derivation
Confidence-interval coverage in a finite model · qf-mod49-ci-interpretation-boundary · candidate
Canonical question JSONL · English JSONL · Validation report