Direct MSE without the decomposition formula

Goal and scope

Weight losses directly at every theta, leaving decomposition for later.

A complete finite scalar model, not MSE inferred from one loss.

Givens and symbols

• work

• G1 — Theta={1/4,3/4}, Omega={00,01,10,11}; P_(1/4)=(9,3,3,1)/16 and P_(3/4)=(1,3,3,9)/16.
• G2 — U(x)=(x1+x2)/4 has total table(0,1/4,1/4,1/2) in Omega order, with target g(theta)=theta.

• θ — fixed unknown parameter
• U — one total statistic

Rules and sources

• mko_definition · cited — Each theta requires a complete normalized finite sampling law. Compute sum over listed t of P_theta(T=t)*(t−theta)^2 directly, without using the later bias-variance decomposition. A multi-theta table keeps each law separate. MSE0 is equivalent to T=theta almost surely under P_theta. One observed estimate/loss does not define the expected loss, though its value may coincide. Unbiasedness does not imply zero MSE, and MSE is neither automatically variance nor a consistency certificate. This slice uses scalar parameters and exact rational laws. · mko-estimator-mean-squared-error
• work · prerequisite — Finite laws have nonnegative probabilities summing to one; a statistic depends only on data, and expectation sums all weighted outcomes. MSEtheta(U)=sum_u Ptheta(U=u)(u-theta)²; a nonnegative finite sum is zero exactly when all positive-weight terms vanish.

Main derivation

U\in\{0,1/4,1/2\}

P_{1/4}^U=(9,6,1)/16,\quad P_{3/4}^U=(1,6,9)/16

\theta=1/4,\quad(U-\theta)^2

(1/16,0,1/16)

\operatorname{MSE}_{1/4}(U)=\sum_uP_{1/4}(U=u)(u-1/4)^2

(9/16)(1/16)+(6/16)0+(1/16)(1/16)=5/128

\theta=3/4,\quad(U-\theta)^2

(9/16,1/4,1/16)

\operatorname{MSE}_{3/4}(U)

(1/16)(9/16)+(6/16)(1/4)+(9/16)(1/16)=21/128

\theta=1/4,\quad x=01

(U(01)-\theta)^2=0\ne5/128

\operatorname{MSE}_\theta(U)=0

(u-\theta)^2=0\text{ at every positive-mass }u

(u-\theta)^2=0\iff u=\theta

\operatorname{MSE}_\theta(U)=0\iff U=\theta\quad P_\theta\text{-a.s.}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, G2, S07, work — Both MSEs here are positive, and no consistency claim follows.
• S07 · premise_audit · G1, G2, S06, work — At a listed sample point of probability mass zero, even a finite nonzero loss contributes 0 to the MSE weighted sum.
• S06 · premise_audit · G1, G2, S05, work — The distinction is numerical, not merely terminological.
• S05 · premise_audit · G1, G2, S04, work — No averaging across theta replaces pointwise MSE.
• S04 · premise_audit · G1, G2, S03, work — The squared difference between1/4 and 3/4 is 1/4.
• S03 · premise_audit · G1, G2, S02, work — The middle mass remains represented despite zero loss.
• S02 · premise_audit · G1, G2, S01, work — The target theta is not the mean 1/8.
• S01 · premise_audit · G1, G2, work — Losses and weights must use the same theta.

Error controls and negative controls

Direct route

List all three losses without invoking later decomposition.

Parameter alignment

At theta=3/4 pair losses(9/16,1/4,1/16) with the second law.

• The loss at 01 is zero, so MSE is zero. → reject — The first full-law MSE is 5/128.
• MSE targets E[U] rather than theta. → reject — That would be variance; the declared target is theta.
• Zero MSE requires hitting even at null atoms. → reject — The sum constrains positive weights only, giving an a.s. equality.

Result and limitations

Direct calculation gives MSE_(1/4)=5/128 and MSE_(3/4)=21/128 without decomposition.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 6b8dc75670c80b7c314c315d9ba0f05efd2bc67d8ee79d74ae4c39ae8edb70e6
• original source SHA-256: 5ba358558d2e86f84d7bddf9c595239e23b51aee26b3b923b284b2a70e7901b4
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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