# Question 25163: MSE Is the Complete Expected Squared Error under One Theta Law [Return to catalogue page 126](https://ocme.evemisslab.com/en/questions/page/126/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations/#q-20261015-advance-20261114-r05-estimator-mean-squared-error-0013) 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. R0049-25163 · q-20261015-advance-20261114-r05-estimator-mean-squared-error-0013 ## Original question and givens Use scalar theta=0, complete listed estimator values=[-3, 0, 3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate). - G001 — Use scalar theta=0, complete listed estimator values=[-3, 0, 3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, 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-estimator-mean-squared-error](https://ocme.evemisslab.com/en/mko/mko-estimator-mean-squared-error/) ## Step-by-step derivation S001 · impliesBefore source roles = [mse-single-theta-exact] After derived = [[theta, weighted_error_rows, MSE, certificate]] Conditions: none List the full source field order without JavaScript.LP boundary modes all have6 fields but select different roles from the original mode,not answer shape. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore MSE ORIGINAL single theta ORIGINAL probability cumulative0 := sum(0, 1/4) After MSE ORIGINAL single theta ORIGINAL probability cumulative0 = 1/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore MSE ORIGINAL single theta ORIGINAL probability cumulative1 := sum(1/4, 1/2) After MSE ORIGINAL single theta ORIGINAL probability cumulative1 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore MSE ORIGINAL single theta ORIGINAL probability cumulative2 := sum(3/4, 1/4) After MSE ORIGINAL single theta ORIGINAL probability cumulative2 = 1 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore source roles = [MSE ORIGINAL single theta, 0, [-3, 0, 3], [[1, 4], [1, 2], [1, 4]]] After derived = [[[-3, 1/4], [0, 1/2], [3, 1/4]], 1] Conditions: none Fix this same P_theta.Check original-order nonnegative masses summing to1,then sort values numerically with their aligned masses.Keep null rows,without renormalizing or treating estimator values as parameter indices. Step identifiers and dependenciesRule: MSE_same_theta_complete_law_numeric_sorted_ALIGNED_null_rows · Uses: G001, S004 S006 · calculationBefore MSE ORIGINAL single theta exact mean product0 := mul(-3, 1/4) After MSE ORIGINAL single theta exact mean product0 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore MSE ORIGINAL single theta exact mean cumulative0 := sum(0, -3/4) After MSE ORIGINAL single theta exact mean cumulative0 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore MSE ORIGINAL single theta exact mean product1 := mul(0, 1/2) After MSE ORIGINAL single theta exact mean product1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore MSE ORIGINAL single theta exact mean cumulative1 := sum(-3/4, 0) After MSE ORIGINAL single theta exact mean cumulative1 = -3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore MSE ORIGINAL single theta exact mean product2 := mul(3, 1/4) After MSE ORIGINAL single theta exact mean product2 = 3/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore MSE ORIGINAL single theta exact mean cumulative2 := sum(-3/4, 3/4) After MSE ORIGINAL single theta exact mean cumulative2 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore MSE ORIGINAL single theta error row0 := sub(-3, 0) After MSE ORIGINAL single theta error row0 = -3 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore MSE ORIGINAL single theta squared error row0 := mul(-3, -3) After MSE ORIGINAL single theta squared error row0 = 9 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore MSE ORIGINAL single theta weighted error row0 := mul(1/4, 9) After MSE ORIGINAL single theta weighted error row0 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore MSE ORIGINAL single theta MSE cumulative0 := sum(0, 9/4) After MSE ORIGINAL single theta MSE cumulative0 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore MSE ORIGINAL single theta error row1 := sub(0, 0) After MSE ORIGINAL single theta error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore MSE ORIGINAL single theta squared error row1 := mul(0, 0) After MSE ORIGINAL single theta squared error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore MSE ORIGINAL single theta weighted error row1 := mul(1/2, 0) After MSE ORIGINAL single theta weighted error row1 = 0 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore MSE ORIGINAL single theta MSE cumulative1 := sum(9/4, 0) After MSE ORIGINAL single theta MSE cumulative1 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore MSE ORIGINAL single theta error row2 := sub(3, 0) After MSE ORIGINAL single theta error row2 = 3 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore MSE ORIGINAL single theta squared error row2 := mul(3, 3) After MSE ORIGINAL single theta squared error row2 = 9 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore MSE ORIGINAL single theta weighted error row2 := mul(1/4, 9) After MSE ORIGINAL single theta weighted error row2 = 9/4 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore MSE ORIGINAL single theta MSE cumulative2 := sum(9/4, 9/4) After MSE ORIGINAL single theta MSE cumulative2 = 9/2 Conditions: none Substitute original roles and established intermediates exactly,without rounding or substituting a single observation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · impliesBefore source roles = [MSE ORIGINAL single theta, 0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]]] After derived = [0, 9/2, No, No, Yes] Conditions: none MSE directly sums all weighted squared errors,without a bias-variance decomposition.Nonnegative terms sum to0 iff the error vanishes at every positive-mass value;null values may differ.Both sides False still give a True iff. Step identifiers and dependenciesRule: MSE_nonnegative_terms_zero_IFF_POSITIVE_mass_equality · Uses: G001, S023 ## Reverse checks - S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S023 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row2 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative1 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row1 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row1 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative0 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row0 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative2 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative1 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative0 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute MSE ORIGINAL single theta ORIGINAL probability cumulative0 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 (0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition) 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: 7e315e7001180b2c8a773d22079316e5f937f60674e5c769342087b6e91c9e5f - candidate record SHA-256: e4c5a0ef0674015ba9b6ef5571235c7a9e3e54a83278d34efc451fc7dc43d38f - semantic review receipt SHA-256: 6212b8b0b3b273f3a514e26ebd17d7e81219c56a8f47e40047ee2b6bf01d1ba3 - question-first witness SHA-256: f32a23d5e1b66c7794c60def58741bcdfacff24f3cdfc9bc48109c8f71543cf3 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0013.json)