# Question 27122: Average Coverage, Posterior Probability and Selection-Adjusted Coverage Are Different Quantities [Return to catalogue page 136](https://ocme.evemisslab.com/en/questions/page/136/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261118-r03-finite-confidence-interval-coverage-foundations/#q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0097) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0053-27122 · q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0097 ## Original question and givens 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). - G001 — 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). ## 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. - [mko-finite-confidence-interval-coverage](https://ocme.evemisslab.com/en/mko/mko-finite-confidence-interval-coverage/) ## Step-by-step derivation S001 · impliesBefore 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. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · impliesBefore source roles = [[0, 1], [a, b], [[[22, 24], [2, 24]], [[2, 24], [22, 24]]], [[0, 0], [1, 1]]] After derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[11/12, 1/12], [1/12, 11/12]], [[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 dependenciesRule: CI_ORIGINAL_theta_sample_interval_probability_alignment_Source13_numeric · Uses: G001, S001 S003 · calculationBefore CI theta0 ORIGINAL probability normalization column0 := sum(0, 11/12) After CI theta0 ORIGINAL probability normalization column0 = 11/12 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore CI theta0 ORIGINAL probability normalization column1 := sum(11/12, 1/12) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore CI theta1 ORIGINAL probability normalization column0 := sum(0, 1/12) After CI theta1 ORIGINAL probability normalization column0 = 1/12 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore CI theta1 ORIGINAL probability normalization column1 := sum(1/12, 11/12) 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore CI theta0 ORIGINAL included coverage column0 := sum(0, 11/12) After CI theta0 ORIGINAL included coverage column0 = 11/12 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [0, [a, b], [[0, 0], [1, 1]], [11/12, 1/12]] After derived = [[[a, 0, 0, 0, Yes, 11/12], [b, 1, 1, 0, No, 1/12]], 11/12] 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 dependenciesRule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S007 S009 · calculationBefore CI theta1 ORIGINAL included coverage column1 := sum(0, 11/12) After CI theta1 ORIGINAL included coverage column1 = 11/12 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore source roles = [1, [a, b], [[0, 0], [1, 1]], [1/12, 11/12]] After derived = [[[a, 0, 0, 1, No, 1/12], [b, 1, 1, 1, Yes, 11/12]], 11/12] 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 dependenciesRule: CI_EVERY_sample_closed_endpoint_membership_AND_weighted_coverage · Uses: G001, S009 S011 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore 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. Step identifiers and dependenciesRule: CI_ORIGINAL_theta_weight_pairs_NOT_slot_substitution · Uses: G001, S012 S014 · calculationBefore CI ORIGINAL Bayes numerator theta0 := mul(1/2, 1/12) After CI ORIGINAL Bayes numerator theta0 = 1/24 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore CI ORIGINAL evidence probability running sum0 := sum(0, 1/24) After CI ORIGINAL evidence probability running sum0 = 1/24 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore CI ORIGINAL Bayes numerator theta1 := mul(1/2, 11/12) After CI ORIGINAL Bayes numerator theta1 = 11/24 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore CI ORIGINAL evidence probability running sum1 := sum(1/24, 11/24) 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. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore CI ORIGINAL posterior theta0 := div(1/24, 1/2) After CI ORIGINAL posterior theta0 = 1/12 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore CI ORIGINAL posterior theta1 := div(11/24, 1/2) After CI ORIGINAL posterior theta1 = 11/12 Conditions: none Substitute original roles and established intermediates exactly;no approximation,slot substitution or internally consistent changed operand. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore source roles = [b, [1/2, 1/2], [1/24, 11/24], 1/2] After derived = [[1/12, 11/12], [[0, 11/12], [1, 11/12]], 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. Step identifiers and dependenciesRule: CI_posterior_GIVEN_observed_data_NOT_frequentist_coverage · Uses: G001, S019 ## Reverse checks - 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. ## Result (posterior_confusion, b, [1/12, 11/12], [[0, 11/12], [1, 11/12]], 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. ## 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: eda73db6e99f322488f729f977d2aa1615a41ecd4a4fc9e7cfaf989e5f05b768 - candidate record SHA-256: 718b29813f600bcd304203396b486b46b50bf82237cc906e347dcbbb199e6339 - semantic review receipt SHA-256: 1b7a62514347b82518305d5feb4369a52d7d44c442a0f2b131e9eb3cd7b45f62 - question-first witness SHA-256: 79dc2a1077081f3ea2a03f75e32778163bf05e814cdf4c29ed86e567295f9ee1 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261118-r03-finite-confidence-interval-coverage-0097.json)