# Question 25587: MSE Equals Variance plus Squared Bias under One Complete Theta Law [Return to catalogue page 128](https://ocme.evemisslab.com/en/questions/page/128/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261115-r03-bias-variance-decomposition-foundations/#q-20261015-advance-20261115-r03-bias-variance-decomposition-0012) 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. R0050-25587 · q-20261015-advance-20261115-r03-bias-variance-decomposition-0012 ## Original question and givens Use scalar theta=-1, complete estimator values=[-1, 2, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate). - G001 — Use scalar theta=-1, complete estimator values=[-1, 2, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate). ## 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-bias-variance-decomposition](https://ocme.evemisslab.com/en/mko/mko-bias-variance-decomposition/) ## Step-by-step derivation S001 · impliesBefore source roles = [biasvar-single-theta-identity] After derived = [[theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate]] Conditions: none List the full source field order without JavaScript.LP weak uses13 fields,zero/strong15;premises and roles follow the original operation/mode without padding. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore Biasvar ORIGINAL single theta ORIGINAL mass sum := sum(1/4, 1/2, 1/4) After Biasvar ORIGINAL single theta ORIGINAL mass sum = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore source roles = [Biasvar ORIGINAL single theta, -1, [-1, 2, 5], [[1, 4], [1, 2], [1, 4]]] After derived = [[(-1, 1/4), (2, 1/2), (5, 1/4)], 1] Conditions: none Qualify this same complete P_theta law,with nonnegative original-order mass summing to1,and sort values numerically with their masses.Keep null rows.Unlike the earlier MSE node,this node explicitly proves the decomposition. Step identifiers and dependenciesRule: Biasvar_complete_SAME_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S002 S004 · calculationBefore Biasvar ORIGINAL single theta mean product0 := mul(-1, 1/4) After Biasvar ORIGINAL single theta mean product0 = -1/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Biasvar ORIGINAL single theta mean product1 := mul(2, 1/2) After Biasvar ORIGINAL single theta mean product1 = 1 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Biasvar ORIGINAL single theta mean product2 := mul(5, 1/4) After Biasvar ORIGINAL single theta mean product2 = 5/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Biasvar ORIGINAL single theta mean sum := sum(-1/4, 1, 5/4) After Biasvar ORIGINAL single theta mean sum = 2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Biasvar ORIGINAL single theta bias mean minus theta := sub(2, -1) After Biasvar ORIGINAL single theta bias mean minus theta = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Biasvar ORIGINAL single theta centered deviation0 := sub(-1, 2) After Biasvar ORIGINAL single theta centered deviation0 = -3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Biasvar ORIGINAL single theta centered square0 := mul(-3, -3) After Biasvar ORIGINAL single theta centered square0 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Biasvar ORIGINAL single theta variance weighted term0 := mul(1/4, 9) After Biasvar ORIGINAL single theta variance weighted term0 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Biasvar ORIGINAL single theta centered deviation1 := sub(2, 2) After Biasvar ORIGINAL single theta centered deviation1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Biasvar ORIGINAL single theta centered square1 := mul(0, 0) After Biasvar ORIGINAL single theta centered square1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Biasvar ORIGINAL single theta variance weighted term1 := mul(1/2, 0) After Biasvar ORIGINAL single theta variance weighted term1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Biasvar ORIGINAL single theta centered deviation2 := sub(5, 2) After Biasvar ORIGINAL single theta centered deviation2 = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Biasvar ORIGINAL single theta centered square2 := mul(3, 3) After Biasvar ORIGINAL single theta centered square2 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Biasvar ORIGINAL single theta variance weighted term2 := mul(1/4, 9) After Biasvar ORIGINAL single theta variance weighted term2 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Biasvar ORIGINAL single theta variance sum := sum(9/4, 0, 9/4) After Biasvar ORIGINAL single theta variance sum = 9/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Biasvar ORIGINAL single theta theta error0 := sub(-1, -1) After Biasvar ORIGINAL single theta theta error0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore Biasvar ORIGINAL single theta theta square0 := mul(0, 0) After Biasvar ORIGINAL single theta theta square0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Biasvar ORIGINAL single theta direct MSE term0 := mul(1/4, 0) After Biasvar ORIGINAL single theta direct MSE term0 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Biasvar ORIGINAL single theta theta error1 := sub(2, -1) After Biasvar ORIGINAL single theta theta error1 = 3 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Biasvar ORIGINAL single theta theta square1 := mul(3, 3) After Biasvar ORIGINAL single theta theta square1 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Biasvar ORIGINAL single theta direct MSE term1 := mul(1/2, 9) After Biasvar ORIGINAL single theta direct MSE term1 = 9/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Biasvar ORIGINAL single theta theta error2 := sub(5, -1) After Biasvar ORIGINAL single theta theta error2 = 6 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore Biasvar ORIGINAL single theta theta square2 := mul(6, 6) After Biasvar ORIGINAL single theta theta square2 = 36 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore Biasvar ORIGINAL single theta direct MSE term2 := mul(1/4, 36) After Biasvar ORIGINAL single theta direct MSE term2 = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore Biasvar ORIGINAL single theta direct MSE sum := sum(0, 9/2, 9) After Biasvar ORIGINAL single theta direct MSE sum = 27/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · calculationBefore Biasvar ORIGINAL single theta centered mean product0 := mul(1/4, -3) After Biasvar ORIGINAL single theta centered mean product0 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S028 S030 · calculationBefore Biasvar ORIGINAL single theta centered mean product1 := mul(1/2, 0) After Biasvar ORIGINAL single theta centered mean product1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore Biasvar ORIGINAL single theta centered mean product2 := mul(1/4, 3) After Biasvar ORIGINAL single theta centered mean product2 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore Biasvar ORIGINAL single theta centered mean sum := sum(-3/4, 0, 3/4) After Biasvar ORIGINAL single theta centered mean sum = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore Biasvar ORIGINAL single theta bias times centered mean := mul(3, 0) After Biasvar ORIGINAL single theta bias times centered mean = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore Biasvar ORIGINAL single theta cross twice inner := mul(2, 0) After Biasvar ORIGINAL single theta cross twice inner = 0 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore Biasvar ORIGINAL single theta squared bias := mul(3, 3) After Biasvar ORIGINAL single theta squared bias = 9 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore Biasvar ORIGINAL single theta variance plus squared bias := sum(9/2, 9) After Biasvar ORIGINAL single theta variance plus squared bias = 27/2 Conditions: none Substitute original roles and established intermediates exactly,without floats;extra teaching identities do not impose new source numeric restrictions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · impliesBefore source roles = [Biasvar ORIGINAL single theta, -1, 2, 3, [(-1, 1/4), (2, 1/2), (5, 1/4)]] After derived = [[[-1, 1/4, -3, 3, 0, 9, -18, 9, 0], [2, 1/2, 0, 3, 3, 0, 0, 9, 9], [5, 1/4, 3, 3, 6, 9, 18, 9, 36]], 0, 0, 9/2, 27/2, 27/2, Yes] Conditions: none For every value,T−theta=(T−mean)+bias.Square to centered²+2*bias*centered+bias² and weight under the same law.E[T−mean]=E[T]−mean*1=0 cancels the cross term,without requiring bias0 or every deviation0.Total mass1 leaves bias².Pointwise cross terms are extra exact teaching values,not new source-profile restrictions. Step identifiers and dependenciesRule: Biasvar_EVERY_value_square_expansion_and_GENERAL_cross_cancellation · Uses: G001, S036 ## Reverse checks - S037 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance plus squared bias with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta squared bias with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta cross twice inner with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta bias times centered mean with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean sum with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean product2 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean product1 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered mean product0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE sum with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE term2 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta square2 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta error2 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE term1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta square1 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta error1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta direct MSE term0 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta square0 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta theta error0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance sum with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance weighted term2 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered square2 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered deviation2 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance weighted term1 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered square1 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered deviation1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta variance weighted term0 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered square0 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta centered deviation0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta bias mean minus theta with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean sum with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean product2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean product1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta mean product0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute Biasvar ORIGINAL single theta ORIGINAL mass sum with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (-1, [[-1, 1/4], [2, 1/2], [5, 1/4]], 2, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law) 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: e1f9006ba4576b6e36aafbb0374c6d1316cc6dc0787894457b1ad56a802216a0 - candidate record SHA-256: 50172e274823721799a57a1f4989b3259e32fcd8ccdcd6659aa1293592443fa3 - semantic review receipt SHA-256: 74c63a2c12285a3b2073f8d9f102bc2ba5fb7ed890c5b7c3b8afa94f032eda82 - question-first witness SHA-256: 6d56428610f8e86c45fd00a2ad3b27ad9c052956946f0b91d71a68536f79d95f - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261115-r03-bias-variance-decomposition-0012.json)