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R0053-27114 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0089
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).
G001 — 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).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.
Before
source roles = [ci-interpretation-boundary]
After
derived = [[mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_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.
Rule: source_ordered_fields_without_JS · Uses: G001
Before
source roles = [[0, 1], [a, b], [[[14, 16], [2, 16]], [[2, 16], [14, 16]]], [[0, 0], [1, 1]]]
After
derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[7/8, 1/8], [1/8, 7/8]], [[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.
Rule: CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · Uses: G001, S001
Before
CI theta0 ORIGINAL probability normalization column0 := sum(0, 7/8)
After
CI theta0 ORIGINAL probability normalization column0 = 7/8
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S002
Before
CI theta0 ORIGINAL probability normalization column1 := sum(7/8, 1/8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S003
Before
CI theta1 ORIGINAL probability normalization column0 := sum(0, 1/8)
After
CI theta1 ORIGINAL probability normalization column0 = 1/8
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S004
Before
CI theta1 ORIGINAL probability normalization column1 := sum(1/8, 7/8)
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.
Rule: exact_rational_arithmetic · Uses: G001, S005
Before
CI theta0 ORIGINAL included coverage column0 := sum(0, 7/8)
After
CI theta0 ORIGINAL included coverage column0 = 7/8
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S006
Before
source roles = [0, [a, b], [[0, 0], [1, 1]], [7/8, 1/8]]
After
derived = [[[a, 0, 0, 0, Yes, 7/8], [b, 1, 1, 0, No, 1/8]], 7/8]
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.
Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S007
Before
CI theta1 ORIGINAL included coverage column1 := sum(0, 7/8)
After
CI theta1 ORIGINAL included coverage column1 = 7/8
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S008
Before
source roles = [1, [a, b], [[0, 0], [1, 1]], [1/8, 7/8]]
After
derived = [[[a, 0, 0, 1, No, 1/8], [b, 1, 1, 1, Yes, 7/8]], 7/8]
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.
Rule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S009
Before
CI prior ORIGINAL normalized weight0 := sum(0, 1/2)
After
CI prior ORIGINAL normalized weight0 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S010
Before
CI prior ORIGINAL normalized weight1 := sum(1/2, 1/2)
After
CI prior ORIGINAL normalized weight1 = 1
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S011
Before
source roles = [CI prior, [0, 1], [[2, 4], [2, 4]]]
After
derived = [[(0, 1/2), (1, 1/2)], 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.
Rule: CI_ORIGINAL_theta_weight_pairs_NOT_slot_substitution · Uses: G001, S012
Before
CI ORIGINAL Bayes numerator theta0 := mul(1/2, 1/8)
After
CI ORIGINAL Bayes numerator theta0 = 1/16
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S013
Before
CI ORIGINAL evidence probability running sum0 := sum(0, 1/16)
After
CI ORIGINAL evidence probability running sum0 = 1/16
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S014
Before
CI ORIGINAL Bayes numerator theta1 := mul(1/2, 7/8)
After
CI ORIGINAL Bayes numerator theta1 = 7/16
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S015
Before
CI ORIGINAL evidence probability running sum1 := sum(1/16, 7/16)
After
CI ORIGINAL evidence probability running sum1 = 1/2
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S016
Before
CI ORIGINAL posterior theta0 := div(1/16, 1/2)
After
CI ORIGINAL posterior theta0 = 1/8
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S017
Before
CI ORIGINAL posterior theta1 := div(7/16, 1/2)
After
CI ORIGINAL posterior theta1 = 7/8
Conditions: none
Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand.
Rule: exact_rational_arithmetic · Uses: G001, S018
Before
source roles = [b, [1/2, 1/2], [1/16, 7/16], 1/2]
After
derived = [[1/8, 7/8], [[0, 7/8], [1, 7/8]], No]
Conditions: none
Posterior=prior(theta)*P_theta(observed)/positive evidence.It treats theta as random and conditions on observed data.Coverage fixes theta and randomizes the sample,so these probabilities/conditions are not interchangeable.
Rule: CI_posterior_GIVEN_observed_data_NOT_frequentist_coverage · Uses: G001, S019
S020 · 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.S019 · premise_and_result_audit — Recompute CI ORIGINAL posterior theta1 with exact rational arithmetic, not rounded decimals.S018 · premise_and_result_audit — Recompute CI ORIGINAL posterior theta0 with exact rational arithmetic, not rounded decimals.S017 · premise_and_result_audit — Recompute CI ORIGINAL evidence probability running sum1 with exact rational arithmetic, not rounded decimals.S016 · premise_and_result_audit — Recompute CI ORIGINAL Bayes numerator theta1 with exact rational arithmetic, not rounded decimals.S015 · premise_and_result_audit — Recompute CI ORIGINAL evidence probability running sum0 with exact rational arithmetic, not rounded decimals.S014 · premise_and_result_audit — Recompute CI ORIGINAL Bayes numerator theta0 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 prior ORIGINAL normalized weight1 with exact rational arithmetic, not rounded decimals.S011 · premise_and_result_audit — Recompute CI prior 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 — 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.(posterior_confusion, b, [1/8, 7/8], [[0, 7/8], [1, 7/8]], No, posterior_given_observed_data_is_not_frequentist_coverage)
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 428b75f30b5692c9f8547db5dcaa8edefd878e22f231b19490941b6c17b9aed6candidate record SHA-256: fa22b20647f23c73cdc831302b5916c055b53021e2d033c7c57cefba73f03959semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1Lean: NOT_RUN · human review: NOT_PERFORMED