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

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This is an owner-adopted natural-language teaching guide. Finite/arithmetic checks are not universal proofs; no Lean was run, and review scope is recorded per revision. Original source status labels are retained as historical data. Formulas retain exact TeX; no new semantic AST is invented.

derivation-mko-finite-maximum-likelihood-estimator-01 · ocme-new142-reader-adoption-20261021



## 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

S01 · impliesBefore

P_0,P_1,P_2

After

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

Conditions: none

Rule: work · Uses: G1, work

Leaf basis: work

All three rows are nonnegative laws on the five-point carrier including z.



S02 · impliesBefore

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

After

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

Conditions: none

Rule: work · Uses: G1, S01, work

Leaf basis: work

Compare all three parameter likelihoods; the unique maximum is at 0.



S03 · impliesBefore

x=b

After

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

Conditions: none

Rule: work · Uses: G1, S02, work

Leaf basis: work

The second observation uniquely maximizes at 1.



S04 · impliesBefore

x=c

After

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

Conditions: none

Rule: work · Uses: G1, S03, work

Leaf basis: work

Compare all three values for the third observation; the maximizer is 2.



S05 · impliesBefore

x=d

After

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

Conditions: none

Rule: work · Uses: G1, S04, work

Leaf basis: work

d has positive mass in every law and supplies a genuine tie witness.



S06 · impliesBefore

x=z

After

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

Conditions: none

Rule: work · Uses: G1, S05, work

Leaf basis: work

In the all-zero column every parameter attains maximum0, giving the full argmax set.



S07 · impliesBefore

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

After

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

Conditions: none

Rule: work · Uses: G1, S06, work

Leaf basis: work

The fixed order selects2 for both ties and completes all samples.



S08 · impliesBefore

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

After

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

Conditions: none

Rule: work · Uses: G1, S07, work

Leaf basis: work

At every outcome the selected value belongs to its full argmax.







## 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 argmaxCompare all three parameters0,1,2 for every sample.

Two tiesDistinguish 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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