第 25566 題：偏誤－變異數分解

原題與已知條件

逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={"laws":[[3,[3,4,5],[[1,4],[1,2],[1,4]]],[4,[3,4,5],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 逐theta完整finite sampling law上的exact mean、bias、variance與MSE。操作=biasvar-multi-theta-table；資料={"laws":[[3,[3,4,5],[[1,4],[1,2],[1,4]]],[4,[3,4,5],[[1,4],[1,2],[1,4]]]]}。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-bias-variance-decomposition

逐步推導

source roles = [biasvar-multi-theta-table]

derived = [[theta_identity_rows, all_identities_hold, certificate]]

Biasvar ORIGINAL law index0 ORIGINAL mass sum := sum(1/4, 1/2, 1/4)

Biasvar ORIGINAL law index0 ORIGINAL mass sum = 1

source roles = [Biasvar ORIGINAL law index0, 3, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]]

derived = [[(3, 1/4), (4, 1/2), (5, 1/4)], 1]

Biasvar ORIGINAL law index0 mean product0 := mul(3, 1/4)

Biasvar ORIGINAL law index0 mean product0 = 3/4

Biasvar ORIGINAL law index0 mean product1 := mul(4, 1/2)

Biasvar ORIGINAL law index0 mean product1 = 2

Biasvar ORIGINAL law index0 mean product2 := mul(5, 1/4)

Biasvar ORIGINAL law index0 mean product2 = 5/4

Biasvar ORIGINAL law index0 mean sum := sum(3/4, 2, 5/4)

Biasvar ORIGINAL law index0 mean sum = 4

Biasvar ORIGINAL law index0 bias mean minus theta := sub(4, 3)

Biasvar ORIGINAL law index0 bias mean minus theta = 1

Biasvar ORIGINAL law index0 centered deviation0 := sub(3, 4)

Biasvar ORIGINAL law index0 centered deviation0 = -1

Biasvar ORIGINAL law index0 centered square0 := mul(-1, -1)

Biasvar ORIGINAL law index0 centered square0 = 1

Biasvar ORIGINAL law index0 variance weighted term0 := mul(1/4, 1)

Biasvar ORIGINAL law index0 variance weighted term0 = 1/4

Biasvar ORIGINAL law index0 centered deviation1 := sub(4, 4)

Biasvar ORIGINAL law index0 centered deviation1 = 0

Biasvar ORIGINAL law index0 centered square1 := mul(0, 0)

Biasvar ORIGINAL law index0 centered square1 = 0

Biasvar ORIGINAL law index0 variance weighted term1 := mul(1/2, 0)

Biasvar ORIGINAL law index0 variance weighted term1 = 0

Biasvar ORIGINAL law index0 centered deviation2 := sub(5, 4)

Biasvar ORIGINAL law index0 centered deviation2 = 1

Biasvar ORIGINAL law index0 centered square2 := mul(1, 1)

Biasvar ORIGINAL law index0 centered square2 = 1

Biasvar ORIGINAL law index0 variance weighted term2 := mul(1/4, 1)

Biasvar ORIGINAL law index0 variance weighted term2 = 1/4

Biasvar ORIGINAL law index0 variance sum := sum(1/4, 0, 1/4)

Biasvar ORIGINAL law index0 variance sum = 1/2

Biasvar ORIGINAL law index0 theta error0 := sub(3, 3)

Biasvar ORIGINAL law index0 theta error0 = 0

Biasvar ORIGINAL law index0 theta square0 := mul(0, 0)

Biasvar ORIGINAL law index0 theta square0 = 0

Biasvar ORIGINAL law index0 direct MSE term0 := mul(1/4, 0)

Biasvar ORIGINAL law index0 direct MSE term0 = 0

Biasvar ORIGINAL law index0 theta error1 := sub(4, 3)

Biasvar ORIGINAL law index0 theta error1 = 1

Biasvar ORIGINAL law index0 theta square1 := mul(1, 1)

Biasvar ORIGINAL law index0 theta square1 = 1

Biasvar ORIGINAL law index0 direct MSE term1 := mul(1/2, 1)

Biasvar ORIGINAL law index0 direct MSE term1 = 1/2

Biasvar ORIGINAL law index0 theta error2 := sub(5, 3)

Biasvar ORIGINAL law index0 theta error2 = 2

Biasvar ORIGINAL law index0 theta square2 := mul(2, 2)

Biasvar ORIGINAL law index0 theta square2 = 4

Biasvar ORIGINAL law index0 direct MSE term2 := mul(1/4, 4)

Biasvar ORIGINAL law index0 direct MSE term2 = 1

Biasvar ORIGINAL law index0 direct MSE sum := sum(0, 1/2, 1)

Biasvar ORIGINAL law index0 direct MSE sum = 3/2

Biasvar ORIGINAL law index0 centered mean product0 := mul(1/4, -1)

Biasvar ORIGINAL law index0 centered mean product0 = -1/4

Biasvar ORIGINAL law index0 centered mean product1 := mul(1/2, 0)

Biasvar ORIGINAL law index0 centered mean product1 = 0

Biasvar ORIGINAL law index0 centered mean product2 := mul(1/4, 1)

Biasvar ORIGINAL law index0 centered mean product2 = 1/4

Biasvar ORIGINAL law index0 centered mean sum := sum(-1/4, 0, 1/4)

Biasvar ORIGINAL law index0 centered mean sum = 0

Biasvar ORIGINAL law index0 bias times centered mean := mul(1, 0)

Biasvar ORIGINAL law index0 bias times centered mean = 0

Biasvar ORIGINAL law index0 cross twice inner := mul(2, 0)

Biasvar ORIGINAL law index0 cross twice inner = 0

Biasvar ORIGINAL law index0 squared bias := mul(1, 1)

Biasvar ORIGINAL law index0 squared bias = 1

Biasvar ORIGINAL law index0 variance plus squared bias := sum(1/2, 1)

Biasvar ORIGINAL law index0 variance plus squared bias = 3/2

source roles = [Biasvar ORIGINAL law index0, 3, 4, 1, [(3, 1/4), (4, 1/2), (5, 1/4)]]

derived = [[[3, 1/4, -1, 1, 0, 1, -2, 1, 0], [4, 1/2, 0, 1, 1, 0, 0, 1, 1], [5, 1/4, 1, 1, 2, 1, 2, 1, 4]], 0, 0, 1/2, 3/2, 3/2, 是]

Biasvar ORIGINAL law index1 ORIGINAL mass sum := sum(1/4, 1/2, 1/4)

Biasvar ORIGINAL law index1 ORIGINAL mass sum = 1

source roles = [Biasvar ORIGINAL law index1, 4, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]]

derived = [[(3, 1/4), (4, 1/2), (5, 1/4)], 1]

Biasvar ORIGINAL law index1 mean product0 := mul(3, 1/4)

Biasvar ORIGINAL law index1 mean product0 = 3/4

Biasvar ORIGINAL law index1 mean product1 := mul(4, 1/2)

Biasvar ORIGINAL law index1 mean product1 = 2

Biasvar ORIGINAL law index1 mean product2 := mul(5, 1/4)

Biasvar ORIGINAL law index1 mean product2 = 5/4

Biasvar ORIGINAL law index1 mean sum := sum(3/4, 2, 5/4)

Biasvar ORIGINAL law index1 mean sum = 4

Biasvar ORIGINAL law index1 bias mean minus theta := sub(4, 4)

Biasvar ORIGINAL law index1 bias mean minus theta = 0

Biasvar ORIGINAL law index1 centered deviation0 := sub(3, 4)

Biasvar ORIGINAL law index1 centered deviation0 = -1

Biasvar ORIGINAL law index1 centered square0 := mul(-1, -1)

Biasvar ORIGINAL law index1 centered square0 = 1

Biasvar ORIGINAL law index1 variance weighted term0 := mul(1/4, 1)

Biasvar ORIGINAL law index1 variance weighted term0 = 1/4

Biasvar ORIGINAL law index1 centered deviation1 := sub(4, 4)

Biasvar ORIGINAL law index1 centered deviation1 = 0

Biasvar ORIGINAL law index1 centered square1 := mul(0, 0)

Biasvar ORIGINAL law index1 centered square1 = 0

Biasvar ORIGINAL law index1 variance weighted term1 := mul(1/2, 0)

Biasvar ORIGINAL law index1 variance weighted term1 = 0

Biasvar ORIGINAL law index1 centered deviation2 := sub(5, 4)

Biasvar ORIGINAL law index1 centered deviation2 = 1

Biasvar ORIGINAL law index1 centered square2 := mul(1, 1)

Biasvar ORIGINAL law index1 centered square2 = 1

Biasvar ORIGINAL law index1 variance weighted term2 := mul(1/4, 1)

Biasvar ORIGINAL law index1 variance weighted term2 = 1/4

Biasvar ORIGINAL law index1 variance sum := sum(1/4, 0, 1/4)

Biasvar ORIGINAL law index1 variance sum = 1/2

Biasvar ORIGINAL law index1 theta error0 := sub(3, 4)

Biasvar ORIGINAL law index1 theta error0 = -1

Biasvar ORIGINAL law index1 theta square0 := mul(-1, -1)

Biasvar ORIGINAL law index1 theta square0 = 1

Biasvar ORIGINAL law index1 direct MSE term0 := mul(1/4, 1)

Biasvar ORIGINAL law index1 direct MSE term0 = 1/4

Biasvar ORIGINAL law index1 theta error1 := sub(4, 4)

Biasvar ORIGINAL law index1 theta error1 = 0

Biasvar ORIGINAL law index1 theta square1 := mul(0, 0)

Biasvar ORIGINAL law index1 theta square1 = 0

Biasvar ORIGINAL law index1 direct MSE term1 := mul(1/2, 0)

Biasvar ORIGINAL law index1 direct MSE term1 = 0

Biasvar ORIGINAL law index1 theta error2 := sub(5, 4)

Biasvar ORIGINAL law index1 theta error2 = 1

Biasvar ORIGINAL law index1 theta square2 := mul(1, 1)

Biasvar ORIGINAL law index1 theta square2 = 1

Biasvar ORIGINAL law index1 direct MSE term2 := mul(1/4, 1)

Biasvar ORIGINAL law index1 direct MSE term2 = 1/4

Biasvar ORIGINAL law index1 direct MSE sum := sum(1/4, 0, 1/4)

Biasvar ORIGINAL law index1 direct MSE sum = 1/2

Biasvar ORIGINAL law index1 centered mean product0 := mul(1/4, -1)

Biasvar ORIGINAL law index1 centered mean product0 = -1/4

Biasvar ORIGINAL law index1 centered mean product1 := mul(1/2, 0)

Biasvar ORIGINAL law index1 centered mean product1 = 0

Biasvar ORIGINAL law index1 centered mean product2 := mul(1/4, 1)

Biasvar ORIGINAL law index1 centered mean product2 = 1/4

Biasvar ORIGINAL law index1 centered mean sum := sum(-1/4, 0, 1/4)

Biasvar ORIGINAL law index1 centered mean sum = 0

Biasvar ORIGINAL law index1 bias times centered mean := mul(0, 0)

Biasvar ORIGINAL law index1 bias times centered mean = 0

Biasvar ORIGINAL law index1 cross twice inner := mul(2, 0)

Biasvar ORIGINAL law index1 cross twice inner = 0

Biasvar ORIGINAL law index1 squared bias := mul(0, 0)

Biasvar ORIGINAL law index1 squared bias = 0

Biasvar ORIGINAL law index1 variance plus squared bias := sum(1/2, 0)

Biasvar ORIGINAL law index1 variance plus squared bias = 1/2

source roles = [Biasvar ORIGINAL law index1, 4, 4, 0, [(3, 1/4), (4, 1/2), (5, 1/4)]]

derived = [[[3, 1/4, -1, 0, -1, 1, 0, 0, 1], [4, 1/2, 0, 0, 0, 0, 0, 0, 0], [5, 1/4, 1, 0, 1, 1, 0, 0, 1]], 0, 0, 1/2, 1/2, 1/2, 是]

source roles = [[[[3, 1], [[[3, 1], [1, 4]], [[4, 1], [1, 2]], [[5, 1], [1, 4]]]], [[4, 1], [[[3, 1], [1, 4]], [[4, 1], [1, 2]], [[5, 1], [1, 4]]]]]]

derived = [[0, 1], [[3, 4, 1, 1/2, 3/2, 3/2, 是], [4, 4, 0, 1/2, 1/2, 1/2, 是]]]

逐步反向核對

• S074 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S073 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S072 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 variance plus squared bias，不以小數近似替代。
• S071 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 squared bias，不以小數近似替代。
• S070 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 cross twice inner，不以小數近似替代。
• S069 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 bias times centered mean，不以小數近似替代。
• S068 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered mean sum，不以小數近似替代。
• S067 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered mean product2，不以小數近似替代。
• S066 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered mean product1，不以小數近似替代。
• S065 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered mean product0，不以小數近似替代。
• S064 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 direct MSE sum，不以小數近似替代。
• S063 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 direct MSE term2，不以小數近似替代。
• S062 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 theta square2，不以小數近似替代。
• S061 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 theta error2，不以小數近似替代。
• S060 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 direct MSE term1，不以小數近似替代。
• S059 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 theta square1，不以小數近似替代。
• S058 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 theta error1，不以小數近似替代。
• S057 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 direct MSE term0，不以小數近似替代。
• S056 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 theta square0，不以小數近似替代。
• S055 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 theta error0，不以小數近似替代。
• S054 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 variance sum，不以小數近似替代。
• S053 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 variance weighted term2，不以小數近似替代。
• S052 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered square2，不以小數近似替代。
• S051 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered deviation2，不以小數近似替代。
• S050 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 variance weighted term1，不以小數近似替代。
• S049 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered square1，不以小數近似替代。
• S048 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered deviation1，不以小數近似替代。
• S047 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 variance weighted term0，不以小數近似替代。
• S046 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered square0，不以小數近似替代。
• S045 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 centered deviation0，不以小數近似替代。
• S044 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 bias mean minus theta，不以小數近似替代。
• S043 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 mean sum，不以小數近似替代。
• S042 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 mean product2，不以小數近似替代。
• S041 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 mean product1，不以小數近似替代。
• S040 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 mean product0，不以小數近似替代。
• S039 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S038 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index1 ORIGINAL mass sum，不以小數近似替代。
• S037 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S036 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 variance plus squared bias，不以小數近似替代。
• S035 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 squared bias，不以小數近似替代。
• S034 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 cross twice inner，不以小數近似替代。
• S033 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 bias times centered mean，不以小數近似替代。
• S032 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered mean sum，不以小數近似替代。
• S031 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered mean product2，不以小數近似替代。
• S030 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered mean product1，不以小數近似替代。
• S029 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered mean product0，不以小數近似替代。
• S028 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 direct MSE sum，不以小數近似替代。
• S027 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 direct MSE term2，不以小數近似替代。
• S026 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 theta square2，不以小數近似替代。
• S025 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 theta error2，不以小數近似替代。
• S024 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 direct MSE term1，不以小數近似替代。
• S023 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 theta square1，不以小數近似替代。
• S022 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 theta error1，不以小數近似替代。
• S021 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 direct MSE term0，不以小數近似替代。
• S020 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 theta square0，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 theta error0，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 variance sum，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 variance weighted term2，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered square2，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered deviation2，不以小數近似替代。
• S014 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 variance weighted term1，不以小數近似替代。
• S013 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered square1，不以小數近似替代。
• S012 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered deviation1，不以小數近似替代。
• S011 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 variance weighted term0，不以小數近似替代。
• S010 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered square0，不以小數近似替代。
• S009 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 centered deviation0，不以小數近似替代。
• S008 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 bias mean minus theta，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 mean sum，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 mean product2，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 mean product1，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 mean product0，不以小數近似替代。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 以精確有理數重算 Biasvar ORIGINAL law index0 ORIGINAL mass sum，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([[3, 4, 1, 1/2, 3/2, 3/2, 是], [4, 4, 0, 1/2, 1/2, 1/2, 是]], 是, identity_is_pointwise_in_theta_not_an_asymptotic_claim)

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。