MLE with a positive-mass tie and an all-zero likelihood column

Goal and scope

Compute every argmax and a total estimator using a fixed tie rule.

A finite exact model; no unbiasedness, consistency or posterior inference.

Givens and symbols

• work

• G1 — Theta={0,1,2}, Omega={a,b,c,d,z}. Rows are(1/2,1/8,1/8,1/4,0),(1/8,1/2,1/8,1/4,0),(1/8,1/8,1/2,1/4,0). Fix tie priority2,0,1 independently of unknown theta.

• A(x) — full argmax set
• hat θ — tie-broken MLE

Rules and sources

• mko_definition · cited — Every theta has a normalized finite law on the complete observation carrier. For an observed x construct every parameter likelihood and the full argmax set. A fixed deterministic parameter-independent tie order gives a total estimator on all observations. Ties and all-zero likelihoods need explicit witnesses. MLE maximizes likelihood, not posterior probability; uniqueness, unbiasedness, consistency and asymptotic optimality do not follow automatically. · mko-finite-maximum-likelihood-estimator
• work · prerequisite — Likelihood is L(theta;x)=Ptheta(x); argmax contains all maximizing parameters; a fixed full parameter permutation breaks ties.

Main derivation

P_0,P_1,P_2

\sum_xP_j(x)=1/2+1/8+1/8+1/4=1\quad(j=0,1,2)

x=a,\quad L(\theta;a)

(L(0;a),L(1;a),L(2;a))=(1/2,1/8,1/8),\quad A(a)=\{0\}

x=b

L(\theta;b)=(1/8,1/2,1/8),\quad A(b)=\{1\}

x=c

L(\theta;c)=(1/8,1/8,1/2),\quad A(c)=\{2\}

x=d

L(\theta;d)=(1/4,1/4,1/4),\quad A(d)=\{0,1,2\}

x=z

L(\theta;z)=(0,0,0),\quad A(z)=\{0,1,2\}

\widehat\theta(x)=\text{first element of }A(x)\text{ in }(2,0,1)

(\widehat\theta(a),\widehat\theta(b),\widehat\theta(c),\widehat\theta(d),\widehat\theta(z))=(0,1,2,2,2)

\forall x\in\Omega:\ \widehat\theta(x)\in A(x)

\widehat\theta\text{ is a total finite MLE}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — Totality includes impossible sample z without asserting an observed-data probability there.
• S07 · premise_audit · G1, S06, work — The tie order is precommitted and cannot inspect true theta.
• S06 · premise_audit · G1, S05, work — Do not normalize0/0 into a posterior.
• S05 · premise_audit · G1, S04, work — Do not assume a unique MLE.
• S04 · premise_audit · G1, S03, work — The c case has been independently covered.
• S03 · premise_audit · G1, S02, work — Do not reuse parameter0 from the a case.
• S02 · premise_audit · G1, S01, work — Argmax is a parameter set, not the maximum value 1/2.
• S01 · premise_audit · G1, work — A zero-mass sample still belongs to the estimator's total domain.

Error controls and negative controls

Full argmax

Compare all three parameters0,1,2 for every sample.

Two ties

Distinguish the positive-likelihood tie at d from the all-zero tie at z.

• The all-zero column has no maximizer. → reject — All three parameters attain maximum0.
• At d, the likelihood has the unique maximizing parameter 2. → reject — Argmax has three values;2 is selected only by the tie order.
• Use true theta to prioritize ties. → reject — The estimator rule cannot depend on unknown theta.

Result and limitations

At d and z the argmax is Theta; the fixed order gives total table(0,1,2,2,2).

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: b70f7d0e33dd447f7ea1ecc0d95b7e12332b17dc0445dcd95ea962c56a1233e5
• original source SHA-256: fc9738f97fb73ca2a370a55d0540a368591e138d436c52e5271233f12285e7d6
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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