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Question 27111: Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0053-27111 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0086

Original question and givens

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).

  • G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [ci-interpretation-boundary]

After

derived = [[mode, selected_samples, unconditional_coverage_table, selected_mass_and_conditional_coverage_by_theta, unchanged_coverage_without_recomputation_claim, boundary]]

Conditions: none

List the complete source field order and mode-specific roles without JavaScript or answer-shape inference;No,undefined and infinity retain their distinct meanings.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · implies

Before

source roles = [[0, 1], [a, b], [[[11, 13], [2, 13]], [[2, 13], [11, 13]]], [[0, 0], [1, 1]]]

After

derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[11/13, 2/13], [2/13, 11/13]], [[a, [0, 0], 0], [b, [1, 1], 1]]]

Conditions: none

Bind intervals to ORIGINAL samples and probability rows/columns to ORIGINAL theta/samples BEFORE sorting those data together.Source13 theta order is numeric and sample order English canonical.Intervals are ordered and closed;each theta law is nonnegative and normalized.The same precommitted,total I(X) does not depend on unknown theta.

Step identifiers and dependencies

Rule: CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · Uses: G001, S001

S003 · calculation

Before

CI theta0 ORIGINAL probability normalization column0 := sum(0, 11/13)

After

CI theta0 ORIGINAL probability normalization column0 = 11/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

CI theta0 ORIGINAL probability normalization column1 := sum(11/13, 2/13)

After

CI theta0 ORIGINAL probability normalization column1 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

CI theta1 ORIGINAL probability normalization column0 := sum(0, 2/13)

After

CI theta1 ORIGINAL probability normalization column0 = 2/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

CI theta1 ORIGINAL probability normalization column1 := sum(2/13, 11/13)

After

CI theta1 ORIGINAL probability normalization column1 = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

CI theta0 ORIGINAL included coverage column0 := sum(0, 11/13)

After

CI theta0 ORIGINAL included coverage column0 = 11/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

source roles = [0, [a, b], [[0, 0], [1, 1]], [11/13, 2/13]]

After

derived = [[[a, 0, 0, 0, Yes, 11/13], [b, 1, 1, 0, No, 2/13]], 11/13]

Conditions: none

Fix the true theta and test EVERY possible sample for L(x)<=theta<=U(x),including equality at either endpoint.Sum P_theta(x) only over hits.This is repeated-sampling coverage,not a posterior for theta after seeing data.

Step identifiers and dependencies

Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S007

S009 · calculation

Before

CI theta1 ORIGINAL included coverage column1 := sum(0, 11/13)

After

CI theta1 ORIGINAL included coverage column1 = 11/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

source roles = [1, [a, b], [[0, 0], [1, 1]], [2/13, 11/13]]

After

derived = [[[a, 0, 0, 1, No, 2/13], [b, 1, 1, 1, Yes, 11/13]], 11/13]

Conditions: none

Fix the true theta and test EVERY possible sample for L(x)<=theta<=U(x),including equality at either endpoint.Sum P_theta(x) only over hits.This is repeated-sampling coverage,not a posterior for theta after seeing data.

Step identifiers and dependencies

Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S009

S011 · calculation

Before

CI theta0 ORIGINAL selected mass column0 := sum(0, 11/13)

After

CI theta0 ORIGINAL selected mass column0 = 11/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

CI theta0 ORIGINAL selected hit column0 := sum(0, 11/13)

After

CI theta0 ORIGINAL selected hit column0 = 11/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

CI theta0 ORIGINAL selected conditional coverage := div(11/13, 11/13)

After

CI theta0 ORIGINAL selected conditional coverage = 1

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

source roles = [0, [a], 11/13, 11/13]

After

derived = [1]

Conditions: none

First compute selection mass;divide hit by mass only when mass>0.At zero mass retain literal undefined,not0 or0/0.Selection can be valid when included in the procedure;the error is reusing coverage without recomputation,not that every selection is invalid.

Step identifiers and dependencies

Rule: CI_EACH_theta_selected_MASS_hit_and_undefined_at_zero · Uses: G001, S013

S015 · calculation

Before

CI theta1 ORIGINAL selected mass column0 := sum(0, 2/13)

After

CI theta1 ORIGINAL selected mass column0 = 2/13

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

CI theta1 ORIGINAL selected conditional coverage := div(0, 2/13)

After

CI theta1 ORIGINAL selected conditional coverage = 0

Conditions: none

Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [1, [a], 2/13, 0]

After

derived = [0]

Conditions: none

First compute selection mass;divide hit by mass only when mass>0.At zero mass retain literal undefined,not0 or0/0.Selection can be valid when included in the procedure;the error is reusing coverage without recomputation,not that every selection is invalid.

Step identifiers and dependencies

Rule: CI_EACH_theta_selected_MASS_hit_and_undefined_at_zero · Uses: G001, S016

S018 · implies

Before

source roles = [[a], [[0, 11/13], [1, 11/13]]]

After

derived = [[a], [[0, 11/13, 1], [1, 2/13, 0]], No]

Conditions: none

The source treats selected samples extensionally,deduplicating and canonical-sorting them.Legal duplicates are neither new samples nor automatically invalid.Conditional coverage is recomputed for each theta and may coincide with original coverage.

Step identifiers and dependencies

Rule: CI_selected_samples_extensional_dedup_not_automatic_invalidity · Uses: G001, S017

Reverse checks

  • S018 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S017 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S016 · premise_and_result_audit — Recompute CI theta1 ORIGINAL selected conditional coverage with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute CI theta1 ORIGINAL selected mass column0 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S013 · premise_and_result_audit — Recompute CI theta0 ORIGINAL selected conditional coverage with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Recompute CI theta0 ORIGINAL selected hit column0 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute CI theta0 ORIGINAL selected mass column0 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S009 · premise_and_result_audit — Recompute CI theta1 ORIGINAL included coverage column1 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S007 · premise_and_result_audit — Recompute CI theta0 ORIGINAL included coverage column0 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute CI theta1 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute CI theta1 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute CI theta0 ORIGINAL probability normalization column1 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute CI theta0 ORIGINAL probability normalization column0 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S001 · premise_and_result_audit — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.

Result

(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)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: 7f28e4288743b6c08cd433722594e4635fba341489722634698f1d98fc67c452
  • candidate record SHA-256: 9dde4581409bccff0983ce0585264200b3f464372c3baca3e718e31ac140f0ab
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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