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Question 27113: 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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R0053-27113 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0088

Original question and givens

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).

  • G001 — 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).

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, 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]]

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], [[[13, 15], [2, 15]], [[2, 15], [13, 15]]], [[0, 0], [0, 0]]]

After

derived = [[0, 1], [a, b], [[0, 0], [0, 0]], [[13/15, 2/15], [2/15, 13/15]], [[a, [0, 0], 0], [b, [0, 0], 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, 13/15)

After

CI theta0 ORIGINAL probability normalization column0 = 13/15

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(13/15, 2/15)

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, 2/15)

After

CI theta1 ORIGINAL probability normalization column0 = 2/15

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(2/15, 13/15)

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, 13/15)

After

CI theta0 ORIGINAL included coverage column0 = 13/15

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 · calculation

Before

CI theta0 ORIGINAL included coverage column1 := sum(13/15, 2/15)

After

CI theta0 ORIGINAL included coverage 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, S007

S009 · implies

Before

source roles = [0, [a, b], [[0, 0], [0, 0]], [13/15, 2/15]]

After

derived = [[[a, 0, 0, 0, Yes, 13/15], [b, 0, 0, 0, Yes, 2/15]], 1]

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, S008

S010 · implies

Before

source roles = [1, [a, b], [[0, 0], [0, 0]], [2/15, 13/15]]

After

derived = [[[a, 0, 0, 1, No, 2/15], [b, 0, 0, 1, No, 13/15]], 0]

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 average ORIGINAL normalized weight0 := sum(0, 1/3)

After

CI average ORIGINAL normalized weight0 = 1/3

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 average ORIGINAL normalized weight1 := sum(1/3, 2/3)

After

CI average ORIGINAL normalized weight1 = 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, S011

S013 · implies

Before

source roles = [CI average, [0, 1], [[1, 3], [2, 3]]]

After

derived = [[(0, 1/3), (1, 2/3)], 1]

Conditions: none

Bind prior/averaging weights to ORIGINAL theta before numeric theta ordering;move each pair together.Do not substitute a slot for theta or sort weights alone.

Step identifiers and dependencies

Rule: CI_ORIGINAL_theta_weight_pairs_NOT_slot_substitution · Uses: G001, S012

S014 · calculation

Before

CI ORIGINAL coverage times theta weight0 := mul(1, 1/3)

After

CI ORIGINAL coverage times theta weight0 = 1/3

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, S013

S015 · calculation

Before

CI ORIGINAL weighted average running sum0 := sum(0, 1/3)

After

CI ORIGINAL weighted average running sum0 = 1/3

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 ORIGINAL coverage times theta weight1 := mul(0, 2/3)

After

CI ORIGINAL coverage times theta weight1 = 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 · calculation

Before

CI ORIGINAL weighted average running sum1 := sum(1/3, 0)

After

CI ORIGINAL weighted average running sum1 = 1/3

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, S016

S018 · implies

Before

source roles = [[[0, 1], [1, 0]], [1/3, 2/3], 1/3]

After

derived = [1/3, 0, Yes, Yes, Yes, No]

Conditions: none

Require genuinely distinct coverages and an ACTUAL weighted average>=target>minimum,checking every condition.This witnesses average coverage masking uniform failure;identical coverages or no strict gap do not.

Step identifiers and dependencies

Rule: CI_ACTUAL_average_masks_uniform_failure_distinct_strict_witness · 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 — Recompute CI ORIGINAL weighted average running sum1 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute CI ORIGINAL coverage times theta weight1 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute CI ORIGINAL weighted average running sum0 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Recompute CI ORIGINAL coverage times theta weight0 with exact rational arithmetic, not rounded decimals.
  • S013 · 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.
  • S012 · premise_and_result_audit — Recompute CI average ORIGINAL normalized weight1 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute CI average ORIGINAL normalized weight0 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 — Reverse-check the complete original data,step operands and premises;finite checks,conditional named-theorem use and general formal proof remain distinct.
  • S008 · premise_and_result_audit — Recompute CI theta0 ORIGINAL included coverage column1 with exact rational arithmetic, not rounded decimals.
  • 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

(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)

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: d860df4bd53125511f7571891ddd23a2035aab245dd0ebf41ad84c34bb27a0e9
  • candidate record SHA-256: a1b62cadaca3de1ae4f241cecaa0636f6cbf3f596114a9245d9aff61e2da73f4
  • semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62
  • question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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