# Question 576: One Fixed Parameter-Independent Tie Order Produces a Total Finite Estimator

[Return to catalogue page 3](https://ocme.evemisslab.com/en/questions/page/3/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-foundations/#q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0051)

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.

P1000-0576 · q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0051



## Original question and givens

Use complete theta labels=[a0, b0], observations=[x0, y0] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a0, b0] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).


- G001 — Use complete theta labels=[a0, b0], observations=[x0, y0] and aligned exact probability rows=[[[3, 4], [1, 4]], [[1, 4], [3, 4]]]. Each theta row is nonnegative and sums to1. Sort labels with their associated rows/columns, never independently. Tie order=[a0, b0] is one complete parameter permutation, fixed independently of the unknown theta. Give (tie_order, estimator_rows, total_estimator, certificate).





## Definition and scope

Keep each observation's complete argmax, then pick the first member in one fixed parameter-independent tie order, giving a total estimator.


- [mko-finite-maximum-likelihood-estimator](https://ocme.evemisslab.com/en/mko/mko-finite-maximum-likelihood-estimator/)





## Step-by-step derivation

S001 · impliesBefore

Θ=[a0, b0]; Ω=[x0, y0]

After

[[3/4, 1/4], [1/4, 3/4]]

Conditions: none

Each row is one parameter law and each column an outcome; reorder probabilities together with labels.

Step identifiers and dependenciesRule: full_parametric_model · Uses: G001





S002 · calculationBefore

P_a0(Ω) := sum(3/4, 1/4)

After

P_a0(Ω) = 1

Conditions: none

Check this law's complete total mass first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

P_b0(Ω) := sum(1/4, 3/4)

After

P_b0(Ω) = 1

Conditions: none

Check this law's complete total mass first.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

tie priority=[a0, b0]

After

one fixed ordering, independent of the observed outcome

Conditions: none

Declare one fixed tie order, then choose its first member within each complete argmax.

Step identifiers and dependenciesRule: fixed_tie_rule · Uses: G001, S003





S005 · impliesBefore

parameter=a0; observed=x0

After

L(a0;x0)=3/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependenciesRule: fixed_likelihood_lookup · Uses: G001, S004





S006 · impliesBefore

parameter=b0; observed=x0

After

L(b0;x0)=1/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependenciesRule: fixed_likelihood_lookup · Uses: G001, S005





S007 · impliesBefore

[[a0, 3/4], [b0, 1/4]]

After

maximum=3/4; all argmax=[a0]

Conditions: none

Retain every maximizer; for an all-zero column the complete argmax is still all parameters.

Step identifiers and dependenciesRule: complete_argmax · Uses: G001, S006





S008 · impliesBefore

argmax(x0)=[a0]; priority=[a0, b0]

After

estimator(x0)=a0

Conditions: none

The selector is single-valued; the original argmax need not be a singleton.

Step identifiers and dependenciesRule: deterministic_tie_selection · Uses: G001, S007





S009 · impliesBefore

parameter=a0; observed=y0

After

L(a0;y0)=1/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependenciesRule: fixed_likelihood_lookup · Uses: G001, S008





S010 · impliesBefore

parameter=b0; observed=y0

After

L(b0;y0)=3/4

Conditions: none

Fix the observation and compare its column across parameters, without introducing a prior.

Step identifiers and dependenciesRule: fixed_likelihood_lookup · Uses: G001, S009





S011 · impliesBefore

[[a0, 1/4], [b0, 3/4]]

After

maximum=3/4; all argmax=[b0]

Conditions: none

Retain every maximizer; for an all-zero column the complete argmax is still all parameters.

Step identifiers and dependenciesRule: complete_argmax · Uses: G001, S010





S012 · impliesBefore

argmax(y0)=[b0]; priority=[a0, b0]

After

estimator(y0)=b0

Conditions: none

The selector is single-valued; the original argmax need not be a singleton.

Step identifiers and dependenciesRule: deterministic_tie_selection · Uses: G001, S011





S013 · impliesBefore

defined on 2 outcomes

After

total on all 2 listed outcomes

Conditions: none

Every listed outcome has an output, including an outcome null under all laws.

Step identifiers and dependenciesRule: total_estimator · Uses: G001, S012









## Reverse checks


- S013 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S012 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S011 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S010 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S009 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S008 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S007 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S006 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S005 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S004 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.
- S003 · premise_and_result_audit — Recompute P_b0(Ω) with exact rational arithmetic, not rounded decimals.
- S002 · premise_and_result_audit — Recompute P_a0(Ω) with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Retain every parameter and sample label; distinguish null, empty tables, zero distributions and tied optima.





## Result

([a0, b0], [[x0, [a0], a0], [y0, [b0], b0]], Yes, theta_independent_tie_rule_makes_a_total_estimator)

The full substitutions and premise audits recover the original answer, retaining its field order and verdict meanings.





## 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: b0d8bf4423d1d86b10a30f8415c79e9444e1dcf3f8a3558264e03af6a39c6168
- candidate record SHA-256: 4410a81b50d63260717fda0a3a4bbd84a41f8fea6dc63073c2270fdfa9fe7aa4
- semantic review receipt SHA-256: 6c2c6d46aded7582a11ac41a8c0b3bdde2669a53bdb6e796eea2039994a02772
- question-first witness SHA-256: 77e43898df9efdbf8094ad81fe4ab91cf4b4bd82f481eb9b4699b73cb4da612b
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261116-r04-finite-maximum-likelihood-estimator-0051.json)