第 27106 題：有限模型信賴區間覆蓋率

原題與已知條件

完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={"mode":"posterior_confusion","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"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整finite laws與total precommitted theta-independent interval procedure責任。操作=ci-interpretation-boundary；資料={"mode":"posterior_confusion","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"}}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-finite-confidence-interval-coverage

逐步推導

source roles = [ci-interpretation-boundary]

derived = [[mode, observed_sample, posterior_by_theta, pointwise_coverage_table, posterior_equals_coverage_claim, boundary]]

source roles = [[0, 1], [a, b], [[[6, 8], [2, 8]], [[2, 8], [6, 8]]], [[0, 0], [1, 1]]]

derived = [[0, 1], [a, b], [[0, 0], [1, 1]], [[3/4, 1/4], [1/4, 3/4]], [[a, [0, 0], 0], [b, [1, 1], 1]]]

CI theta0 ORIGINAL probability normalization column0 := sum(0, 3/4)

CI theta0 ORIGINAL probability normalization column0 = 3/4

CI theta0 ORIGINAL probability normalization column1 := sum(3/4, 1/4)

CI theta0 ORIGINAL probability normalization column1 = 1

CI theta1 ORIGINAL probability normalization column0 := sum(0, 1/4)

CI theta1 ORIGINAL probability normalization column0 = 1/4

CI theta1 ORIGINAL probability normalization column1 := sum(1/4, 3/4)

CI theta1 ORIGINAL probability normalization column1 = 1

CI theta0 ORIGINAL included coverage column0 := sum(0, 3/4)

CI theta0 ORIGINAL included coverage column0 = 3/4

source roles = [0, [a, b], [[0, 0], [1, 1]], [3/4, 1/4]]

derived = [[[a, 0, 0, 0, 是, 3/4], [b, 1, 1, 0, 否, 1/4]], 3/4]

CI theta1 ORIGINAL included coverage column1 := sum(0, 3/4)

CI theta1 ORIGINAL included coverage column1 = 3/4

source roles = [1, [a, b], [[0, 0], [1, 1]], [1/4, 3/4]]

derived = [[[a, 0, 0, 1, 否, 1/4], [b, 1, 1, 1, 是, 3/4]], 3/4]

CI prior ORIGINAL normalized weight0 := sum(0, 1/2)

CI prior ORIGINAL normalized weight0 = 1/2

CI prior ORIGINAL normalized weight1 := sum(1/2, 1/2)

CI prior ORIGINAL normalized weight1 = 1

source roles = [CI prior, [0, 1], [[2, 4], [2, 4]]]

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

CI ORIGINAL Bayes numerator theta0 := mul(1/2, 1/4)

CI ORIGINAL Bayes numerator theta0 = 1/8

CI ORIGINAL evidence probability running sum0 := sum(0, 1/8)

CI ORIGINAL evidence probability running sum0 = 1/8

CI ORIGINAL Bayes numerator theta1 := mul(1/2, 3/4)

CI ORIGINAL Bayes numerator theta1 = 3/8

CI ORIGINAL evidence probability running sum1 := sum(1/8, 3/8)

CI ORIGINAL evidence probability running sum1 = 1/2

CI ORIGINAL posterior theta0 := div(1/8, 1/2)

CI ORIGINAL posterior theta0 = 1/4

CI ORIGINAL posterior theta1 := div(3/8, 1/2)

CI ORIGINAL posterior theta1 = 3/4

source roles = [b, [1/2, 1/2], [1/8, 3/8], 1/2]

derived = [[1/4, 3/4], [[0, 3/4], [1, 3/4]], 否]

逐步反向核對

• S020 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S019 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL posterior theta1，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL posterior theta0，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL evidence probability running sum1，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL Bayes numerator theta1，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL evidence probability running sum0，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 CI ORIGINAL Bayes numerator theta0，不以小數近似替代。
• S013 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S012 · premise_and_result_audit — 以精確有理數重算 CI prior ORIGINAL normalized weight1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 CI prior ORIGINAL normalized weight0，不以小數近似替代。
• S010 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S009 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL included coverage column1，不以小數近似替代。
• S008 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S007 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL included coverage column0，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL probability normalization column1，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 CI theta1 ORIGINAL probability normalization column0，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL probability normalization column1，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 CI theta0 ORIGINAL probability normalization column0，不以小數近似替代。
• S002 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。
• S001 · premise_and_result_audit — 回查原始完整資料、步驟操作數與本步前提；有限核對、條件式定理引用與一般形式證明分開。

結論

(posterior_confusion, b, [1/4, 3/4], [[0, 3/4], [1, 3/4]], 否, posterior_given_observed_data_is_not_frequentist_coverage)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。