Question 25176: Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

Original question and givens

Use scalar theta=-12, complete listed estimator values=[-12, -11] and aligned probabilities=[1, 0] 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, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).

• G001 — Use scalar theta=-12, complete listed estimator values=[-12, -11] and aligned probabilities=[1, 0] 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, zero_MSE, almost_surely_equal_theta, equivalence_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-estimator-mean-squared-error

Step-by-step derivation

source roles = [mse-zero-iff-almost-sure]

derived = [[theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate]]

MSE ORIGINAL single theta ORIGINAL probability cumulative0 := sum(0, 1)

MSE ORIGINAL single theta ORIGINAL probability cumulative0 = 1

MSE ORIGINAL single theta ORIGINAL probability cumulative1 := sum(1, 0)

MSE ORIGINAL single theta ORIGINAL probability cumulative1 = 1

source roles = [MSE ORIGINAL single theta, -12, [-12, -11], [1, 0]]

derived = [[[-12, 1], [-11, 0]], 1]

MSE ORIGINAL single theta exact mean product0 := mul(-12, 1)

MSE ORIGINAL single theta exact mean product0 = -12

MSE ORIGINAL single theta exact mean cumulative0 := sum(0, -12)

MSE ORIGINAL single theta exact mean cumulative0 = -12

MSE ORIGINAL single theta exact mean product1 := mul(-11, 0)

MSE ORIGINAL single theta exact mean product1 = 0

MSE ORIGINAL single theta exact mean cumulative1 := sum(-12, 0)

MSE ORIGINAL single theta exact mean cumulative1 = -12

MSE ORIGINAL single theta error row0 := sub(-12, -12)

MSE ORIGINAL single theta error row0 = 0

MSE ORIGINAL single theta squared error row0 := mul(0, 0)

MSE ORIGINAL single theta squared error row0 = 0

MSE ORIGINAL single theta weighted error row0 := mul(1, 0)

MSE ORIGINAL single theta weighted error row0 = 0

MSE ORIGINAL single theta MSE cumulative0 := sum(0, 0)

MSE ORIGINAL single theta MSE cumulative0 = 0

MSE ORIGINAL single theta error row1 := sub(-11, -12)

MSE ORIGINAL single theta error row1 = 1

MSE ORIGINAL single theta squared error row1 := mul(1, 1)

MSE ORIGINAL single theta squared error row1 = 1

MSE ORIGINAL single theta weighted error row1 := mul(0, 1)

MSE ORIGINAL single theta weighted error row1 = 0

MSE ORIGINAL single theta MSE cumulative1 := sum(0, 0)

MSE ORIGINAL single theta MSE cumulative1 = 0

source roles = [MSE ORIGINAL single theta, -12, [[-12, 1, 0, 0], [-11, 0, 1, 0]]]

derived = [-12, 0, Yes, Yes, Yes]

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative1 with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row1 with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row1 with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row1 with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute MSE ORIGINAL single theta MSE cumulative0 with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute MSE ORIGINAL single theta weighted error row0 with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute MSE ORIGINAL single theta squared error row0 with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute MSE ORIGINAL single theta error row0 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative1 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product1 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean cumulative0 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute MSE ORIGINAL single theta exact mean product0 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

(-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)

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.